@Grivusangel -- Yeah, I was thinking I wouldn't want to use anything on my barbecue grill that wasn't food safe to start with. Vinegar and baking soda sound perfect. If the mixture is good enough to clean my cat's litterbox without me worrying about toxic materials on his paws, then it's good enough for my barbecue grill for sure.
I've also seen on the cooking shows that running an oiled paper towel over the hot grill before cooking helps keep the food from sticking, which also facilitates a much quicker clean-up. I thought that was a great idea, too.