I love Fukuoka. In particular you might like to visit the Genko Historical Museum, which houses both Japanese and Mongolian arms and armour from the 13th century.
The Fukuoka city museum has some great exhibits from the region's history. Fukuoka also contains Japan's very first Zen temple.
More info here:
Fukuoka Travel Guide.
If you have time, I really reccommend heading down to Kumamoto. The castle is well worth the journey. Only the donjon is a concrete reconstruction - the other buildings are wooden. It's a genuinely beautiful building in gorgeous surrounds.
From Kumamoto you can also visit Mount Aso.
More info here:
Kumamoto Travel Guide