Kyushu is quite highly regarded amongst other Asian tourists. Quite a lot to see and do. Be sure to go during mid-may, just after the golden week holidays. The end of May can start to become quite humid and hot (for northerners). There is nothing wrong with renting a car as the island is quite compact. The train system is also quite extensive. If you would like to splurge look into the 7 Star Trans-Kyushu train.
http://www.jrkyushu.co.jp/english/cruisetrain/