Hi,
I think Kyushu would make an excellent pick, and it is easy to get from Fukuoka to many other parts of the island by bullet train. Many just use it as a jumping off point, but there are some very good places there in addition to great food. Nagasaki is also a must-see place, but more isolated and not the best choice as a base. From there you could see Unzen, as well as Sasebo for the nice [island cruise] as well as Huis Ten Bosch. Travel by bullet train is quick and easy, and Kumamoto is not far away. They are still rebuilding from the bad earthquake they had, but there is still lots to explore. A lot of the castle was damaged, but the Suizenji Garden is still there and picturesque. Down at the end of the line is Kagoshima with its beautiful and very active Sakurajima and Senganen Garden.
And if you like hot springs, you are in the hot spring capital of the country - there are so many good places - Unzen, Kurokawa, Beppu and Yufuin
Shikoku is an often bypassed place but it to has some good sights. Takamatsu, Kochi and Matsuyama are the more interesting cities, especially in Takamatsu there is the Ritsurin Garden which is one of the best in Japan, plus in Kotohira, climbing up to the Konpirasan Shrine is a spiritual pilgrimage and one of Japan's best half day hikes. Kochi has some gems as well, including one of Japan's best limestone caves to explore at Ryugado plus the tremendous view from Mt. Godaisan and its lovely castle. Matsuyama has its famous Dogo hot springs, original surviving castle and some beautiful coastline. Unfortunately though getting around the island can be very slow - there are no bullet trains and less people means fewer train trips.
I think you'll still find 3 trips to Japan are not enough!
Good luck.