I think the dining CAN be done fairly cheaply - we rarely do a sit down that is not a character meal (and now that our daughter is 4 and doesn't get to eat for free we definately will not be going to any buffets that are not character meals - just meals where the kids can split an entree)
We bring our own food into the parks so everyone can eat when they are hungry and we don't have to wait in line for food when it is really busy (but since we are having more fun in the parks, we don't feel like we are depriving ourselves).
We eat breakfast in our room and on the way to the park (especially on early entry mornings) - the kids eat fruit, cereal, yogurt, juice, etc., in the room and then we bring along a bagel or something to munch on and baggie full of cereal for a morning snack.
Whenever we go on a trip anywhere, I always bring bread and peanut butter (and a knife and sandwich bags!) to make sandwiches - other than getting smashed occasionally, they hold up pretty well and are great to tide the kids over until the next meal - plus we can eat them while standing in line. They are so cheap that if we don't end up eating them, it's no big loss just to toss them (which rarely happens).
This allows us to eat more relaxed meals when it is slower and we can usually split 2 meals between the 4 of us (no one goes hungry - if we need more food, we order it, but we never start out with 4 meals because they would just go to waste). It also gives us more $ for popcorn and ice cream!
I try to have "good" snacks in the room - string cheese, yogurt, fruit, carrot sticks, trail mix, etc., that they eat while in the room, but I also know they aren't going to get malnourished in a week and as long as they are not filling up on junk and their energy levels are ok, I don't stress too much over what they eat. I will say this about sit down meals - it is the best chance they have for getting served any vegatables during the trip!
With 3 little ones and a large group to keep up with, counter service may be much more enjoyable and "do-able" than sit down meals, but if you can swing one character meal I doubt you will regret it! Also, a lot of the counter dining at Disney is fun - the kids would much rather go to Pizza Planet in MGM than a fancy sit down meal.