I would stay in Fukuoka/Hakata. I've lived in Kumamoto for 4 years, but Hakata is a better central location to use a JR Pass to get to nearby areas. Kyushu is best appreciated by car once you leave the main lines.
Within Fukuoka/Hakata is a famous pottery on the west side of Ohori Park. It has a step kiln that looks like it operates but I wonder how they use it in the middle of town.
Takatori Pottery
http://www.takatoriyaki.jp/web2/indexe.htmlFukuoka/Hakata is also famous for its non-glazed porcelain dolls. If you ask at the information center in the middle of Hakata Station, they can guide you to places that specialize in them.
Hakata Ningyo
http://www.japanvisitor.com/japanese-culture/hakata-dollsBetter than Arita for pottery, and just a short local/private train ride north of Arita is Imari. From Imari, catch a bus (or take a taxi) to Okawachiyama, a town that has nothing but pottery. It's in a nice rural setting and is worth walking around to see what you discover. This place is reached by an express from Hakata to Arita, then a private line to Imari as the fastest route. You could also take the Shinkansen to Shin Tosu and pick up the express there. I would set aside the whole day for this destination as travel time adds up.
Nabeshima Pottery, Okawachiyama
http://findingfukuoka.com/2011/04/26/okawachiyama/Down to the SE of Fukuoka is an old Tokugawa gold mining town, Hita-shi. If you like old architecture, it's a pretty little town accessed by JR. It has its own pottery, Ontayaki, and the town specializes in making geta.
Hita City
http://www.city.hita.oita.jp/english_top.htmlTo the east of Hakata is Sasaguri Town, with the temple, Nanzoin, and the huge bronze reclining Buddha. The temple itself is worth checking out as a stream and small waterfalls come down through the temple area. A tunnel leads you to the new site of the large Buddha. The Buddha is what got me there, but I was more impressed with the rest of the temple. It's also accessed by JR with no transfers.
Nanzoin Buddha
http://www.japanvisitor.com/japan-temples-shrines/nanzoin-templePhotos:
https://www.flickr.com/search/?q=Nanzoin%20fukuokaSouth of the main city is Yanagawa, a town designed by its daimyo to be flooded if danger approached. As a result, there were many canals built, which you can now tour by boat. The going is slow but it takes you through some old samurai neighborhoods with some old buildings. Many people make it a party ride and bring their own food and drinks. To get there you have to take the Nishitetsu Line from Fukuoka instead of a JR line.
Yanagawa:
http://www.japanvisitor.com/japan-city-guides/city-guide-yanagawaFukuoka/Hakata also has decent hotel rates and many good restaurants which you can find out about in any guide. Most people go there for the yatai food stalls, Ohori Park, the Mongol invasion wall remnants, and Dazaifu to the south. Overlooked are the City Museum, which often has special traveling exhibits, the Folk Museum next to the Kushida Shrine and easy shopping in Tenjin and Canal City. It has big city amenities but isn't so large that it wears you out.
Fukuoka also has some of the best ramen in Japan, and you don't have to go to a yatai to find it. I think the Kinryu (Golden Dragon) shop in front of Fukuoka University is the best, but don't go at lunchtime unless you have time to wait in line.
A bit more on the area in a blog:
http://findingfukuoka.com/If you really wanted to push it, you could visit Nagasaki on a day trip. The expresses are fast enough to get you there and back again. Since you already would have seen the Gembaku Museum in Hiroshima, I would skip the one in Nagasaki, they're almost identical, and spend time around the Dejima reconstructions and Glover Gardens, where foreigners houses from the Meiji were moved (and every castella shop in Nagasaki). Japanese like to go to Huis ten Bosch, which is mainly a reconstructed earthquake-proof Dutch Town with other attractions attached. I found the MC Escher Museum to be the most interesting thing there for my $65 (at the time) entry fee. I think the place is interesting to Asians a long way from Europe.
http://english.huistenbosch.co.jp/pdf/map_english.pdf