If you have a visa that lets you stay in Japan more than 90 days (work, culture, student, spousal, permanent resident), your best bet for saving money on train travel is excursion tickets (shuuyuuken).

Their availability changes monthly, so you have to either consult the monthly JR schedule (available in most bookstores) if you read Japanese or have a Japanese friend do it for you. Basically, a shuuyuuken gives you round trip transportation to your destination along with a given number of days of travel within your destination area. For example, there's usually a Kansai-area ticket for Tokyo residents.

Some of the private railways have excursion tickets as well.

None of these are as good a deal as the JR Pass, but there's not a lot you can do about it.