I've written two, complete LGBT short stories & have sent them to some United States literary magazines such as TriQuarterly, the Chicago Review, the New Yorker, and others. I have been fairly unsuccessful with my attempts, but I believe it is not all bad news. With a few of the magazines I applied to, I received personalized rejection letters from the fiction/short story heads. (I don't, I could just be reading into this) They specifically said that my theme was important, but just wasn't right for them. Am I right in assuming this means that they don't want LGBT work in the literary magazine? Or is this really just a default response. The Chicago Review even went so far as to suggest a feminist magazine for my work. I'm just a little confused on what my next steps are if I want to publish these short stories.
I know that I've seen LGBT literary magazines and magazines that specifically seek LGBT fiction. Your best bet might be to try one of those first. A lot of times it's just about finding the right niche.
Just as there are women's stories suitable for feminist magazine and stories involving women suitable for literary mags, there could also be LGBT stories suitable for genre specific mags and/or suitable for general literary mags. So, I don't think their response meant they don't publish any LGBT stories.
[MENTION=36360]killbill[/MENTION] made a good point. It doesn't mean that those other mags exclude LGBT fiction. But your chances of getting published, especially early in your writing career are going to be higher at one of the mags that specifically seeks your genre and type of work. That applies to all types of stories, not just LGBT.
Also, I don't know much about TriQuarterly or the Chicago Review, but the New Yorker is a big, mainstream magazine with a very large circulation and a long history of publishing some of the finest short fiction written in America. It's VERY hard for a new writer to be accepted there. Most of the stories they publish are by well-established and highly-respected writers. The New Yorker is pretty much the top of the mountain - most writers have a long climb ahead of them before they get published there!