I think that you can talk about feminism without relying on identity politics, but, in the first instance, I'm don't think you should. There are serious problems with the notion that social inequalities along identity lines have been overcome and should not be spoken of -- i.e. the all too familiar "identity blind" discourse that provokes such a knee-jerk reaction whenever one makes a statement about gender, class, race, and so on, insinuating that to speak of such things is to perpetuate them. This tendency leads to what might be called gender-blind sexism, color-blind racism, class-blind classism, and so on. I offer this answer tentatively, and not with any sort of claim to elaborating the subject in detail (because that would mean writing a book), and before continuing should say that a number of authors have worked on this question of gender identity (and its abolishion), namely queer theorists like Judith Butler, Eve Sedgwick, Michel Foucault, among others. The best recent work I know of from a feminist perspective is Linda Zerilli's Feminism and the Abyss of Freedom, which is, interestingly, calling for a return to what is often called second-wave feminism. Also, Hardt and Negri's Commonwealth deals extensively with the question of identity politics in relation to revolutionary politics (just to try to answer the "what kinds of stuff is out there" question and not to pretend that I came up with any of this on my own).
Identity, whether gender, class, race, nationality, etc, is a form of violence, and is implicitly connected with most any kind of oppression you can name. There is good reason to want to do away with identity politics and it seems right that this is the goal of any revolutionary politics. But there is nowhere else to start except identity. The Zapatistas have an interesting notion in this regard, encapsulated in their slogan that demands "not to be who we are, but to become what we want." This is a break with identity politics in that it acknowledges the place from which the revolution sets out, in a shared identity of Chipas, the family, Christianity, other traditions, etc., but this is not the intended end of the struggle. The end, instead, is undefined, and does not drive toward a fixed identity but rather a space in which self-transformation is possible (while recognizing that a struggle must take place for this to happen). In terms of gender, this is different from a feminism that might call for equality with, or recognition from, the oppressor (men). This would be to say that the point of starting out must be as feminists, as women, but that this ought not to be the intended end (as dot says in the comment above about reification). The end of feminism must be to abolish gender altogether (working toward queer theory, perhaps), though it is naive to think that that has been accomplished and so identity politics remain necessary, at least in part. The first task of identity politics must be to make visible the violence of identity, but all too often this politics remains stuck in this task and begins to defend that identity as if it were a kind of property instead of moving on the the second task (and the more important) of a struggle for liberation from identity altogether.
Stepping away from the question a little bit, and to close, this is why I would never say "I am an anarchist" or any such phrase, since I don't think of anarchism as an identity but rather as a revolutionary tendency of thought and action that seeks liberation from identity politics (and that's why I like it). Again, I apologize for this all too inadequate answer to a very important question.