I can hardly claim to be fluent, but I do have a rather handy Japanese elctronic dictionary...
This is actually a problem in Japan, too; most of the possible terms are either not neutral, associated with entertainers or sex work, or refer to people based on a surgery they may or may not get.
The problem here, as previously mentioned, is that many of the terms used are derogatory. The main entry for "transsexual" my dictionary gives is 性倒錯者 (seitousakusha, if I'm reading the kanji right); this basically comes out as "person with a sexual abberation/perversion". A slightly less unfortunate term is 性転換者 (seitenkansha), the Japanese equivalent to "person who changes sex". This, of course, basically links your identity to a sex-change operation.
I'm not sure how frequent the use of seitousakusha is, as I had to fiddle with the potential kanji here to get it to match the dictionary's (there are a number of potential words in kanji that fit those basic characters, probably mostly misspelled/not actual words). Seitenkansha popped right up without difficulty.
Newhalf is less depressing; I remember reading a journal article about that one during a research project, though I can't remember what it was called. As I recall, it was created within the transsexual community as a self-descriptive term, so it's not derogatory outright. However, due to the sexualized lines of work of a great many famous newhalf, it carries something of a go-go dancer/sex-worker quality to it. It's also a little old.
I will second the warning against using "futanari"; it is a genre of manga (comic-book style) pornography, roughly akin to "->-bleeped-<-". I'm not even sure if the word is even used to describe actual real-world people...
The problem is that Japan is not really any more accepting of transsexuals than North America is. It's not really seen as something to be upfront about. As such, most people's exposure to them comes from entertainers, who are quite often more stereotypical than the population at large.