Well, when we took our DS to Disney when he was young, our favorite noncharacter meals with him were sit down restaurants. They tend to be a bit quieter. Plus, I liked not having to get up and down to get everyone's food. IMHO, the sit down restaurants have better food than the buffets. Remember that Disney's portions are large, so depending on the kids appetite, you could always share.
That said, we liked the Rose and Crown at Epcot. My DS and I shared a chicken dish that had green beans and I believe a potato side dish. It's hard to find a kid dish without french fries at Disney, so I liked sharing. We had a great table lagoon side to watch Illuminations.
We like Mama Melrose at MGM. It's much quieter and the food is very good. My DS enjoyed watching them make pizzas until our table was ready. They have good kid friendy food such as pizza, etc.
At the Magic Kingdom, I guess outside of a character meal, we really like Pecos Bills for counter service. They have a really good chicken wrap. They have kid friendly hot dogs and mickey cookies. Plus you could share your fries with the little ones.
At Animal Kingdom, I like Tusker House. You could easily get a meal here and split it with the little ones.
I will also say that when my sister in law got sick, and we had to take my DS with us at the last minute to my Birthday dinner at Narcoosses (?sp), they were really great with my DS. He was 4 at the time. They brought his drink with a little umbrella, had a good children's menu (tho. expensive) and had a special dessert made for him. Plus, we saw the fireworks from the MK from our table. It worked out very well (however, it is expensive).
Most importantly. All Disney restaurants will let you split your meals with you kids. You can always order more food if you need it. Also, don't be afraid to ask for half portion of fries/half portion of some veggie with the kids entries (at sit down restaurants). Disney was very accomodating when I made this request.