Jan Dodd is the co-author of The Rough Guide to Vietnam and The Rough Guide to Japan. I would recommend that you try to get out of the cities as much as possible and head for the more rural areas such as Hokkaido, Shikoku, Kyushu and Tohoku. Given that the islands of Japan stretch some 3000km from north to south, the weather patterns vary considerably from one end of the country to another. Honshu's spine of mountains also has an important impact on the local climate. In general, spring and autumn offer the best weather.
Spring tends to be mild, relatively dry and sunny.
Temperatures start to drop and the rains peter out during September, after which there's a bout of dry, sunny weather through to late November. This is the time when everyone flocks to see the glorious autumn colours, especially around Kyoto. It's warmer the further south you go, but even in Kyushu you'll find frost and snow on the higher peaks. Apart from the climate, it's also worth taking note of Japan's national holidays when transport and accommodation is booked up months in advance. If at all possible, avoid travelling during these times.