I recall that in IASFM (yes, Isaac Asimov's Science Fiction Magazine), Asimov once wrote an editorial which covered this subject. Although I don't have that anymore, I think I recall the gist of it pretty well. He noted that a number of science fiction writers over the years had attempted to invent gender-neutral pronouns for the English language, but none had ever gained any traction. Asimov then pointed out that English already had gender-neutral pronouns that work just fine, in the form of "he" and "his". It's rarely difficult to tell from the context when they are being used in a gender-neutral way. The awkward "he or she" construct was a solution to a non-problem.
So, I think I'll stick with Asimov on this. However, I have to admit to being stodgy in my writing habits. I still refer to The Elements of Style (which Asimov also recommended), not to mention Webster's 2nd Edition (the "dord" dictionary), and I still capitalize God (yes, even when He is referred to by pronouns), and I still believe that "flammable" is not a real word and shouldn't be used outside of warning labels that must be understood by semi-literates.