I'm an eastsider so I'm partial to Ellenberger Park in Irvington. From June-September they have the Summer Concert Series, which features local musicians on the first Saturday of every month. This is a great time to bring a picnic.

Good point!

You can check out a pdf guide to the concert series and more arts programming at Indy Parks here. The lunch series would be a nice mid-day plan for picnics, too.

I kinda hate to give this little secret up, but the "arts & nature park" behind the Indianapolis Museum of Art is a great place for a picnic. There is a beautiful little lake back there that has a hiking/biking trail all the way around it, and a meadow that would be perfect place to throw out a blanket. But my personal favorite is to go around to the far side of the lake, where there's just a wedge of land between the lake and the White River, and find a nice spot on the bank for a picnic. You wouldn't believe how nice the White River looks along there. Also, the wildflowers are in full bloom right now, and there is plenty of wildlife. Most notably, a big blue heron that always hangs out around there.

Holcomb Garden on the Butler University campus. Beautiful spot for a picnic, and you can cross one of the bridges and walk in the woods (plenty of paths) on the other side of the canal.