Yes, most of the villages have parks where you'll find benches and can enjoy your own picnic. If you're picnicking, you can also just walk out of the towns in wine country and down the gravel roads among the vineyards until you find a nice place to sit down and enjoy a picnic, remembering however that most are private property and not open for people to walk through.
Nonetheless, I particularly recall Badia a Passignano - seeing it is quite small, there is a small garden on one end as well as a picnic bench along the main road on the other near the cemetery where the view is of all the vineyards that surround the town. Really beautiful.
We've also picnicked (happens when you enjoy hiking ;-) ) near Montefioralle, in Volpaia and Radda - all great spots.