I did look into some places, but because I'm looking to move to a smaller city (in Aichi prefecture) I found that contrary to my expectations, overall it's more EXPENSIVE to live in a smaller city than a bigger city (that is of course if it's still a city--if you go into the countryside it'll probably be a different story).
For cities like Osaka & Tokyo, there are many foreigners concentrated into those two cities, so there are quite a few fully-furnished accommodations to choose from (Sakura House, etc.). And because there are so many, the prices tend to be competitive. Whereas if you go to a smaller neighboring city, there are only one or two companies catering towards foreigners & the prices tend to be quite high for not as many options (the only ones I've been able to find are either completely unfurnished or allow renters to rent furniture to a tune of 5000+ yen per month!).
Another option is to go through the regular route that Japanese residents take, but that opens up another can of worms because then you have to worry about "key money" (it's basically a "present" that you have to give to the landlord that can be up to 6 [or even more] months of rent. It is not a security deposit---at the end of the lease, you do NOT get the money back, so it's a BIG deal. Even if the rent seems reasonable, if you factor that in, it often becomes too expensive and not worth it). Not only that, but you'll need to find someone in Japan who will be willing to be a guarantor. And even if you clear all these things, there is still a bit of discrimination that goes on & there are landlords who won't want to rent to foreigners.