The best season to be there for padding is around July, when the summer monsoon hits. I've never paddled it, but the Yoshino River in Shikoku is supposed to be a nice run http://en.wikipedia.org/wiki/Yoshino_River
You might also check out the Nagara River, in Gifu, which is mellow (Class2) but has (or at least had) some nice play spots. There is a shop the Nagara called Spirit Paddle Shop, run by a guy name Yoshi, who speaks some English. He might be able to give you some tips. http://r156.com/english/
Hokkaido, the north-most island, also has some good rivers and is nice in summer.
In general, the topography is steep, so the river drain quickly after a rain. If you can link up with a Japanese paddler who can help you figure out what rivers will be running, I'd recommend it.
There are a couple of Japanese members of KayakMind who you might contact: