I've done the Antioch Bridge twice, and don't plan to again unless I have a good reason. The "bike" path is a glorified shoulder right next to highway speed traffic, on an incline, so not a very pleasant experience. There are some interesting towns to stop in along River Rd to the north side of the bridge, but the road itself is not very fun and often involves a consistent headwind heading towards Sacramento.
Speaking of bridges, there was an interesting comment made by a Caltrans director while receiving an award at EBBC's Biketopia event last Friday. He pledged that they would not only "make happen" the completion of the Bay Bridge bike path to San Francisco, but also make bike access possible on the Richmond/San Rafael Bridge. There's still lots of unanswered questions, but it was very promising to hear it from the horse's mouth, so to speak. Bike from SF to the North Bay to the East Bay and back? Lots of potential brevet routes, there...
-Robert