It really depends on the activities you want to be doing and which part of Japan you're looking to visit. I've been before in July and still had a great time (I was around Kyoto for most of the trip). While the beginning of June is the start of the rainy season, it doesn't pour down, or even rain, everyday of the month. If you're in the cities or visiting temples I think you would be fine. If you're planning on going to Hokkaido you might not even be affected. This site has some great information on Japan's rainy season that I would agree with from past experience: http://www.japan-guide.com/e/e2277.html
In the end it comes down to where you are, what you're doing and personal preference.
World Nomads over 2 years ago