English already has gender-neutral pronouns: "I," "we," "you," and "they." It's only in the third person singular that a speaker has to keep track of gender. Compared to speakers of many other languages, English speakers have it easy. We don't have to make adjectives agree in gender or number with the nouns they modify, for example, let alone master a maze of honorifics for elders and juniors. And gender mostly becomes a problem for our pronouns only when we're composing a sentence that doesn't refer to individuals but to people in the abstract, as in: "Everybody should be free to vote for the candidate of ... choice."
As Dennis Baron, a professor of English and linguistics at the University of Illinois whose Web of Language blog is in my blogroll, wrote in 1981,
The absence in English of a third-person, common-gender pronoun became apparent when grammarians in the eighteenth century began objecting to the apparently widespread use of they, their, and them with singular, sex-indefinite antecedents on the grounds that it violated number concord...There was always resistance to that approved construction, however, both principled on the ground that generic "he" is sexist, and populist as many if not most people clung to generic "they" for everyday use. At the moment (because dictionaries always change with the changes in language), "the major dictionaries tell us that the plural pronoun they can function as a gender-neutral singular too." In my own writing I boldly waffle, sometimes using "his or her" and variations, sometimes generic "she" (after all, the majority of Americans are female, and majority rules), sometimes "they," sometimes alternating "he" and "she," sometimes invented pronouns from literary sources, and sometimes I just rewrite the sentence to avoid the problem. ("All people should be free to vote for the candidate of their choice." Wait, "choices"?) The issue is unlikely to be resolved in my lifetime, so I don't feel a need either to choose finally or to take a stand on which solution should prevail.
Only [he, the masculine third-person singular pronoun] satisfied the demands of both number concord and style; so despite the fact that it violates gender concord, a requirement the logical-minded prescriptivists were apparently willing to waive, it has become the approved construction.
Something has changed, however, over the past couple of decades. With the increasing visibility of transgender people and people who claim to reject or transcend the gender binary, there have been many calls for new pronouns to refer to them. Many such pronouns have been invented and proposed. This is fine with me, but again, the issue is unlikely to be settled in my lifetime. I'm happy to comply more or less politely with other people's preferred pronouns for themselves. I'm not sure how much difference it makes in practice, since most of the time I'll be using the gender-neutral "you" to address them anyway. Since this is probably true for others as well, I'm not sure how often "Ask me about my pronouns" is going to affect discourse in the real world, except when someone gets written up in the student newspaper and their pronoun will be the main subject of the article.
What I've noticed in most of the coverage I've seen is that most people, whether frothing right-wing gender cops, well-meaning liberals, or frothing left-wing gender cops, don't seem to have noticed that the epicene "they" and the proliferation of supposedly non-binary pronouns are dealing with different phenomena. This well-meaning blog post is typical. For example:
“He,” “she,” or “it” won’t do, “one” doesn’t work when speaking of a specific person, e.g. “Sam washed one’s dishes,” and in some cases even a singular “they” just won’t work – specifically when a name is used, e.g. “Charlie tied their shoes” or “Sam thought they were late to the party.”True enough, but that's because these are different cases, and the writer seems not to realize it. Epicene "they" doesn't refer to "a specific person": it's not really a singular. It's a common-gender pronoun, as Dennis Baron called it. It's not supposed to refer to an individual. (Though sometimes it does with reference to a person of unknown sex/gender, as in "I was chatting with someone on the internet and they said...") On the other hand, if "they" achieved acceptance as the gender-neutral third-person singular pronoun in English, it would be the appropriate pronoun "when a name is used, e.g. 'Charlie tied their shoes'". The real reason why "Sam washed one's dishes" doesn't work is that "one" is a periphrasis for the first person; the sentence would mean that Sam washed my dishes. I also saw an article linked on Facebook which was headlined something like "Every mother should feel free to breastfeed in public if they want to." I can't see any good reason why "she" shouldn't be used in this case, but a common-gender pronoun is the solution to a different problem than the special pronoun being sought for non-binary, agender, or other people.
I've also seen some confusion about some literary examples, like my personal favorite from Marge Piercy's 1976 science-fiction novel Woman on the Edge of Time. In utopian 22nd-century Mouth-of-Mattapoisett, the third-person single pronoun is "person" (for the nominative) and "per" (possessive and accusative). For children, "child" was used, and their species for non-human animals. ("If Tilia [the character's cat] takes a flying leap onto my chest at first dawn from the top of the wardrobe, I get a clear notion that cat is dissatisfied with my conduct.") There was no separate pronoun for persons of ambiguous or unknown gender; from the viewpoint of Connie, the 20th-century visitor to Mattapoisett, most of its citizens were of unknown or ambiguous gender anyway. I think I've seen some references to this novel as a possible model for gender-neutral pronouns, which also seemed to misunderstand what was going on. It wouldn't fit with many people's demand for separate pronouns to reflect their gender identities, and I rather suspect they would reject the one-size-fits-all of person/per.
The same would be true of another 1970s feminist literary experiment, The Cook and the Carpenter by June Arnold under the pseudonym The Carpenter. Arnold substituted na (nominative) and nan (possessive) for "he" and "she" for most of the novel, though she switched to standard pronouns about towards the end. Though I like na and nan, Arnold used them not as proposed changes in English but to stymie readers' attempts to stereotype the characters by sex; hence the reveal toward the end to show who was female and who male. Again, I don't think this alternative would satisfy advocates of separate pronouns for their gender identities. If, as the blogger just cited says, there's a "need for a gender-neutral pronoun" in English, it isn't the same "need" felt by those who want not "gender-neutral" but gender-specific pronouns for their multiplying genders.
There are other possibilities, of course. I seem to have misremembered some details from Samuel Delany's 1984 science-fiction novel Stars in My Pocket like Grains of Sand. What I thought he posited as part of the grammar of one of his planets' languages was that "she" was used for any object of erotic desire, regardless of body configuration; to refer to a person as "she" was to acknowledge that one desired him or her; otherwise the person would be "he." Looking at the text now, I don't think that's what Delany had in mind, but I think it could be a valid grammatical principle. After all, in America when I was growing up there was a general consensus that only women were beautiful or erotically desirable, so that men felt their masculinity was compromised if anyone of whatever gender called them "beautiful." To be desired was to be objectified, a position socially reserved for women. So why not choose pronouns according to the current status of the person you're talking about?
Remember the neutralization of breastfeeding mothers in an Internet magazine: someone pointed out that "she" rather than "they" was appropriate as a generic pronoun, and someone else replied that not all mothers were women, that men could mother too. Granting that, almost no men can breastfeed (though in Woman on the Edge of Time Marge Piercy imagined a future society where we could and did), why not just refer to all mothers as "she," regardless of their plumbing? Why not gender pronouns according to the task being performed by a person at the moment, so that anyone fixing a carburetor would be "he" and anyone changing a diaper would be "she"? Since most of the ostensible nonconformist discourse on gender nowadays is still mired in gender stereotypes and essentialism anyway, why not stop fighting it and go with what people actually think and feel? The idea that "he" and "she" refer to social status rather than body configuration is a tenet of radical feminism (as well as of gay male culture and of traditional male supremacy), so there's no reason why language couldn't reflect it.
Underlying most of this discourse is a naive version of the Sapir-Whorf hypothesis about language, that language determines thought and reflects the values and assumptions of a culture. I think the theory of language created by George Orwell for his novel 1984 is similar: human beings are trapped in the cage of their language, and can't think outside of it. Orwell's account was seriously flawed, if only because language is not that rigid, and human thought is not absolutely controlled by it. (The linguist and pundit John H. McWhorter wrote a short book with a misleading title, trying to refute strict versions of Sapir-Whorf; it's worth reading, but critically.) The fact that people are proposing alternative pronouns to suit their personal politics both disproves the thesis that our thought is absolutely controlled by language (or else they wouldn't be to think about, let alone invent new pronouns) and shows its popularity (because they believe that gender pronouns have thought-controlling power -- if we change them, we change gender norms).
In particular, I'm intrigued by people's evident belief that their gender (conceived as an innate essence) dictates the pronoun that should be used for them, and vice versa. This shows their general ignorance about language and its relation to culture; they barely know their own language and society, never mind others. Some years ago I read an awful book I'm not going to name here, in which the author claimed that because some Native American tribes don't have gender-specific pronouns, they do not have sexist expectations about men and women. I immediately recognized the absurdity of that claim (which I've encountered elsewhere a few times since then). Apart from the fact that many tribes (including the ones he praised) did have sharply-defined gender expectations, I knew that languages like Chinese and Korean don't have gendered pronouns -- yet those societies are extremely sexist and highly gendered. There is a loose, variable and very changeable relation between language and culture, so deliberate changing of pronouns might have some effect on gender norms if it's part of a larger program of social change, but by themselves pronouns don't really have any effect. (The same goes for customs like wives' taking their husbands' name on marriage. They don't do so in China or Japan, yet they still traditionally lose their separate personhood when they marry anyway, and the different custom is not evidence of a lack of sexism in the culture.)
The blogger I cited before mentioned the case of "Ms.", which is probably an exception that proves the rule. It gained currency because so many women liked the idea of a title that wouldn't define them by marital status. In that respect it is more like epicene "they" than it is like "ze." That there weren't a multitude of alternative possibilities, as with pronouns today, was surely a factor in the success of "Ms." But why gender the title (indeed, why have a title) at all? It's as problematic to distinguish between Mr. and Ms. as between "he" and "she," isn't it? Don't we need a gender-neutral alternative to Mr. and Ms. for the non-binary? Apparently at least one has been proposed. I imagine more will follow, as with gender-specific pronouns generally.
Maybe, as with genders, we need a separate third-person pronoun in English for every individual. I say this not to mock the anxieties that drive these proposals, but to point out their confusion and indeed incoherence. Instead of progress, concern with pronouns and multiplying genders seems to me a distraction, spinning our wheels in place. The "need" for new third-person-singular pronouns seems to be psychological rather than grammatical (as is the case with epicene "they"). That doesn't invalidate the felt need, but it needs to be kept in perspective. By all means, let people choose the pronouns and titles they wish, and I'll use them out of courtesy, but I'm not obliged to agree that they represent even a partial or provisional solution to the problem of gender, in language or in the rest of life.