The keyword is ganbanyoku (岩盤浴) in Japanese -- there were a huge boom a few years back, but are now slowly petering out. Here's somebody's experience:

http://www.xenemag.net/e/hot/hot_01.html

And yes, the key to Finnish saunas is the humidity, or more specifically, regulating it yourself by throwing water on the stove. Dry "Finnish" saunas are a crime against humanity... and regrettably common in Japan. (I once saw a warning sign in a sauna in Hokkaido saying that you shouldn't throw water on the stove, because it'll result in hot steam. Well, duh. )