Right now the weekends of July 6-7 and 20-21 look good to me. Sundays on the 14th and 28th also could work.
I've posted this about a half dozen times, but it's worth repeating. There's a great organized KATY Trail 5-day ride starting June 17th. I've done it the last 2 years and plan on doing it yet again in 2013.
http://mostateparks.com/2013-katy-trail-ride