Yoyogi Park in Harajuku near Shibuya almost guaranteed has an event or festival going on during the weekends. Japanese traditional festivals are all well and good, but experiencing more mainstream and modern festivals also provides for a great cultural experience in my opinion.
One festival during the spring I like that is held at Yoyogi Park is called "Haru-kaze", or "Spring-Wind". It's a bit of a Japanese hippie festival with many different bands, performers, and merchants.
People will even set up tents, and bring out their portable grills and ice chests to make it somewhat of a mini-Woodstock (at least as much as can be managed in central Tokyo!)