Every year from 2001 to 2006 the wife and I went to Japan for 2-3.  
Usually to different places (except for Tokyo, our point of entry).  We 
went to Osaka a couple of times and Kyoto twice.  From Nagasaki to 
Morioka, with some little onsen towns like Yudanaka, and any number of 
smaller places like Shiraishi Island, Tonno, Ayu-Wakamatsu, Matsue.  
What fun.

Then financial circumstances, including the disparity between dollar 
and yen precluded it almost 8 years.  Now we're going back.  This time 
for six weeks.  We'd never been to Kanazawa, so we're going there.  
Spending 10 days in Kyoto during which time we'll do a number of 
day-trips. But we still have a few weeks to fill on the dance card, 
roughly between Kobe and the Izu Peninsula.

When we finish in Kyoto and before we leave from Tokyo, we have about 
10 days.  So we were thinking about staying 2 or 3 places along the way 
back.  Maybe Koya-san.  Izu Peninsula.  Nagoya sounds too big after 
Kyoto.  Perhaps snooping through Shizuoka.

Any recommendations for interesting but somewhat out-of-the-way places, 
generally in that part of the country?

All aid appreciated.  I assume many of what passed for regulars over 
the past 12 years are all gone or visit so rarely that they'd not see 
this post.  Ah well!