Sanno Matsuri (Tokyo)
Sanno Matsuri is one of main traditional Festivals of Tokyo that occurs in the early part of the summer, generally just before the rainy season. It dates back to the Edo Era during which the Shogun was the guest of honour at every festival. You will see beautiful shrine processions, bon odori (a type of traditional dance) and lots of festival foods sold at stalls. This festival is held in Asakusa so the traditional setting of the area fits perfectly with the spirit of the matsuri.
Try some staple Japanese festival foods while you are there.
Yosakoi Soran Festival (Sapporo)
There are countless Awa Odori dance festivals in Japan, but none of them is similar to the Yosakoi Soran Festival of Sapporo. Indeed, although the costumes may look the same, the energy and the type of dance performances are quite unique. The festival comes from a fusion of two things: "yosakoi" and "soran". Yosakoi is a type of Japanese dance is inspired by the traditional Awa Odori dance but has been modernized to be more dynamic. As for the "soran" part, it is an ancient type of song and dance that is thought to have been invented by the fishermen of the north. The movements mimic fishermen hauling fish and pulling ropes!
This festival is held in the beautiful city of Sapporo in the north of Japan. June might be a good time to visit the place because it is the only area in Japan that doesn't have the rainy season called Tsuyu during that month.
Hokkaido Shrine Festival
If you'd like to enjoy an important festival with a local vibe, the Hokkaido Shrine Festival, held in Sapporo in June, is an excellent choice. You will get a real feeling of what a matsuri with Shinto roots feels like. Central to the festival is the passing through the city of a procession of Shinto floats called "mikoshi'. People gather along the streets to view the passing of the floats.
This festival marks the start of the summer festival season in Hokkaido so you should see plenty of people wearing yukatas (summer kimono).
If you are in Japan through July, you should also look at this guide of things to do for that month.