There are a ton of untouristy places in Japan, many of which are fascinating. What are your interests though? What do you like about Japan?
Here are some random suggestions:
Okazaki - Okazaki's a lovely little town on the way to Nagoya. It's cherry trees are famous, so I suspect it will be busy during sakura season, but probably with a local crowd.
Inuyama - Lovely town, some really great sights. More off-beat things in the surrounding region.
Shiga Prefecture - The whole prefecture is a gem. There's a ton to see and do in Shiga, and it's beautiful, Japan's largest lake surrounded by hills and mountains. The prefecture sees few tourists, certainly compared to Kyoto. Even Otsu - which is basically on the outskirts of Kyoto - sees far fewer tourists than the city, despite having some of the nicest temples and shrines in the Kyoto area. Given its proximity to Kyoto though, I expect that accommodation in Shiga will also be in high demand during sakura season.
Nara Prefecture - Lots and lots and lots to see and do, in Nara city itself, and also in the surrounding region.
Kobe's good if you like to hike - I think the city's hikes are the best thing about it. Okayama's a nice base, with various interesting attractions in the surrounding region. If you don't mind the extra travel, head to the Sea of Japan. Beautiful coastline in Hyogo (though it's hard to see without a car), sand dunes in Tottori, the impressive Daisen mountain, Matsue, which is one of my favourite little cities in Japan. Matsue-Hiroshima is an easy bus trip, with foreigners getting a steep discount!
You could also head to Shikoku.
Lots and lots of options.