There are three big traditional central city festivals on this weekend
Hot on the heels of the rowdy Sanja matsuri and the shitamachi charms of the Kanda Matsuri comes the relatively refined and subdued Sanno Matsuri (link in Japanese). Based around the hilltop Hie Shrine in Chiyoda Ward (midway between Akasaka and Nagatacho Stations) the festival runs over 10 days from June the 7th to the 17th. While there are plenty of events going on each day, the main procession (shown in the pic above) only happens on even numbered years. The kickoff will be quite low key this weekend, but will climax with a procession on Saturday, June 15th at 11am. The closest stations are Akasaka and Nagatacho.
Just up the road from the Sanno Festival in Yotsuya/Suga is the Yotsuya-Suga Shrine Festival (link in Japanese). The shrine, which dates from the establishment of Edo, There will be lots of stalls selling your typical festival food on both Saturday and Sunday while the main procession with mikoshi (shoulder borne portable shrines) will take place on Saturday. The shrine is about a 10 minute walk from Yotsuya Station.
Another well know festival with more of a shitamachi flavour is the Torigoe Shrine festival happens on both days this weekend. It includes a rowdy procession with the carrying of mikoshi. Closest stations (all within close walking distance) are Asakusa Bashi, Okachimachi and Kuramae.