I'm glad I happened to see this thread!
I'm a second-year JET and a runner. I do a race about every 2 months on average, and I would say my favorite distance right now is the half-marathon, though I did run the Tokyo Marathon this past February (full marathon).
The big websites to look for and sign up for races are:
Sports Entry: http://www.sportsentry.ne.jp/
I would say running is not quite as popular in Japan as it is in the U.S., but I haven't had any problem finding races to run. One thing I noticed is that women hardly ever wear shorts to run -- tights tend to be the popular attire. I've gotten some weird looks running outside in shorts (especially if it's raining or still a little chilly) and I don't even look foreign, but I'd rather be comfortable so I don't care too much.
Something to keep in mind when signing up for races is that the deadline is usually a month or more in advance, even for smaller races. In the U.S., a lot of times there was even day-of-event registration, but I haven't really seen that in Japan. So make sure to plan ahead! Also, if you are a slower runner, make sure to check time limits -- they tend to be a lot stricter here. There may be shorter-distance (3k, 5k) walk/jog oriented events, but I haven't really seen the whole "walking a half marathon" culture you see in the U.S. A 3 hour or 2.5 hour half-marathon limit is pretty normal. Also -- possibly related -- I don't see as many female runners as back in the U.S. Back home, it was usually about half/half, but in Japan, at a typical 10k or half-marathon race I'd see maybe 20 or 25% women.
Running is my biggest hobby -- so ask away if you have any questions!
Toyama CIR, Jan. 2011~ (late upgrade)