The Shops at Worthington Place: We love this cute indoor play area. It's usually clean, well kept and not too busy in the afternoons.
Graeters: The play area inside of Graeters on Bethel Rd. is free, although it's hard to get out of there without buying ice cream!
Nature Center at Highbanks Metro Park: This Nature Center was remodeled a couple of years ago, although the entire center is interesting there is a children's area with a puppet stage, reading library and tree you can go inside.
Eagles Nest: Giant Eagle provides this service for FREE at their Dublin, Hilliard, Grandview and Kingsdale locations. They often have crafts, games, fun toys and movies to entertain the kids (3+) while you shop, grab a coffee or walk laps around the store talking on your phone (ok, I totally admit to doing this)
Center for Creativity at Columbus Museum of Art (Free on Sundays)- The Center for Creativity allows for hands on imaginative and creative play.