The only recent type of "kids eat free" program is that offered to the under 3 crowd. All kids under the age of 3 eat free at any all-you-can eat restaurant - this includes buffets as well as pre-plated meals (e.g. Cinderella's Castle Breakfast) and dinner shows (e.g. Hoop De Doo Revue). If you order food for a child under 3 from the "kid's menu" at any restaurant you will be charged. On our last trip our dd was almost 3 and we took great advantage of this by having as many meals that were free for her as possible.
Some years ago there was a promotion with Disney Club/Magic Kingdom Club where you could get a free kids meal with an adult meal. That was for a relatively limited time and has not been offered for quite some time.