It's festival time in Tokyo - the Kanda Matsuri on the 9th/10th and the Sanja Matsuri in Asakusa
from the 15th-17th are both huge festivals by a major Shinto Shrine in Tokyo.
Don't miss the azaleas and wisteria as mentioned in previous posts; also you'll get to see some roses at the Kyu-Furukawa Gardens (pictured). There is a special night time opening from the 15th until the 24th, when the closing time will be extended to 9pm. Be aware that the popular Ghibli Museum will be closed from the 18th to the 29th.