My preference is to go somewhere that I can mostly get around via JR pass, ideally staying at a place with a good hotel (better than 'good' is a positive but not required) and doing day rail trips. However, I am OK with renting a car for one or two days of the week I'll have for this part of the trip. JR pass is helpful because the mileage award tickets I already have are to/from Tokyo.

As an example of this approach, when I was in Kyushu on my last trip, I took the shinkansen between Fukuoka-Kumamoto (and vice versa) on one day. Then I rented a car to visit Takachiho Gorge.

But there are some really interesting suggestions in the above posts for which this method wouldn't work.