When I first came to Fukuoka in September, I had had a summer of exercising every day, and I was freaking out over the possibility that I would be a lazy bum again. One of the first things I asked the students was where I could find a good, cheap gym. The answer is the Chuuou-ku Taiikukan in Akasaka, about a 10-15 minute walk from GenkiJACS. They have a variety of facilities that I don`t know about too well (I`ve seen people doing kendo, karate, archery, ping-pong, etc) because the main thing I do is just normal gym training. Entry is 260 yen for 2 hours, and you can also rent running shoes there (although the biggest size is 28cm). A locker costs 30 yen. There are a lot of foreigners who train there too. If you can, try to go before the after-work rush.
Besides that gym, there are several other public ones that people at school use, especially Momochi Taiikukan, where some students go to play badminton a few times a week. In the summer, many people go jogging in the huge park Ohori Kouen (although since I don`t have a car I have no idea where I would put my stuff). Other activities people do are jiu-jitsu, bouldering, and hiking. It`s best to ask students around and see what they do and join their activities. Or you can ask your Japanese friends. It`s a great way to meet new people!