I've been on a roadtrip to the Tohoku region (Matsushima, Yamagata, Urabandai, etc.) and in Kyushu (Fukuoka, Sasebo, Nagasaki, Beppu). I think that if I were to compare the price of gas/ tolls/ car rentals vs. train tickets, training would have been cheaper but in terms of time and convenience I think that going by car is the way to go (depending on the area).
I did a rode trip through Kyushu several years ago, hitting Oita, Miyazaki, Kagoshima, Kumamoto and a little little bit of Fukuoka. We also camped out in the car so our costs were quite low. It was a nice way to see lots of places in Kyushu, at our own pace, that we might not have seen at all.