When are the cheapest months to visit Japan?

Cheapest months would be probably the shoulder months between the popular seasons, like February, March, June and November.

Avoid the National holidays for the influx of domestic travels. Specifically, you should choose to avoid these periods if you want to travel cheap:

The first week of May – Golden Week would be the busiest travel period
Mid July and August – here you would have most domestic travel activities because of Obon and school holidays.
Christmas and New Year – busy with both foreign and local tourists
Sakura seasons – April and May. If you really want to catch the Sakura, you may try your luck for early April or late May, the traveling at these times are cheaper and less crowded but there are no guarantee for Sakura blossoms sighting.
Autumn months: August, September and October – may be expensive as it is one of the peak tourist seasons.