There are some very good places that you mention you've been to; perhaps you haven't see all there is.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
For Fukuoka, don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here.
Yamaguchi has some very good places - mostly ignored by the tourist books and guides. But at least there are far less crowds to deal with. One of the very best is the Akiyoshi Limestone Cave, which is part of the Akiyoshidai Quasi-National Park, near Mine City. You can use the pass to take a JR bus from the JR Yamaguchi station. It is truly breath taking. Other good places to see in the area are Hagi Castle in Hagi City, along with its old preserved samurai district, and Mori clan temple grave sites. Shimonoseki has some great places to see as well.
There are many places between or close to Hiroshima. Just a bit past Miyajima is Iwakuni with its iconic historical bridge and mountaintop castle. Onomichi is famous for its temple tour and wildly ornate Kosanji Temple. Nearby Kurashiki is a famous own city for its canals, and Okayama also has its own castle and one of Japan's other Top 3 gardens, Korakuen. If your kids are itching to explore, then in Kagawa there is the Konpirasan Shrine in Kotohira, with a marvelous paved mountain hike.
