According to this article in Salon's website, plastic grocery bags are the worst type of litter in the world. They are choking waterways and killing wildlife. They don't break down, pretty much ever, and they are generally not recycled (our local grocery stores do have bins for them, but you have to remember to bring them it regularly).
It's better to use the cloth totes, or at least the reusable bags made from recycled plastic. Although the latter aren't really washable. You can buy cloth totes at a craft store and decorate them yourself. Or even make your own totes if you have a sewing machine and a bit of talent for that sort of thing.
reusablebags.com has a wide selection of cloth bags, even ones for bagging produce (you can't store the produce in the fridge in them, though).