Sights are uncrowded and accommodation at its cheapest.
The exception is the ski resorts, which are now hitting their stride.
Many businesses close over the New Year period (end of December to early January).
(Jun & Jul, Sep–Nov)
Autumn foliage draws crowds during specific periods in October and November (depending on elevation).
(Apr & May, Aug)
Cherry-blossom season (late March to early April), Golden Week (early May) and O-Bon (mid-August) are peak travel periods, when sights will be crowded and accommodation more expensive (and often fully booked).
Mountain (read: cooler) destinations are most popular in August; this is also the month for many festivals.