Yakushima, Yakushima, Yakushima
If I ever went on the JET Program a second time, my three placement requests would be Yakushima, Yakushima, Yakushima. And no, I'm not just plagiarizing Basho's haiku about Matsushima Japan.
Yakushima, which lies a few hours by boat off of Kyushu and is administratively part of Kagoshima, is not a comfortable place to get to. No direct flights fly from Tokyo as the local airport runway is too short for jet engines. But, this solitude is what makes a trip to Yakushima so incredible. You will meet other travelers hiking the trails and staying in the hostels, but these are genuine folks who went the extra mile to do their research and make it to Japan's heart. Those are the types of people I want to get to know.
I stayed at Yakushima Youth Hostel, and you should, too, as it's minutes from the main port and has a fantastic staff. It's incredible how many folks from Tokyo I meet working in off-the-beaten-path hostels. They all said "no thanks" to salaryman/OL culture and are some of the most satisfied people I've ever met.
In terms of getting around, I hired a guide to take me to Taiko no Iwa. I never hire guides but compared the costs of renting a car, and it made sense financially. Also, I read one too many blogs about folks getting lost and attacked by leeches. It ended up being one of my favorite memories in Japan.
It goes without saying that Japan offers so much more beyond Tokyo and Kyoto. While those are two cities everyone new to Japan should go to, you owe it to yourself to go off the beaten path. You never know what you'll find.