Hi Lauren:
I love Chef Mickey's. If you hadn't already booked that restaurant, it would have been my first recommendation. I'm not sure how old your kids are, or their interests, so I'll run down some great "family fun" restaurants for your to consider, which hopefully will help you find the restaurant experience you are looking for.
My favorite non-character fun Disney restaurant is
Biergarten in the Germany Pavilion at Epcot. As you eat an all-you-can consume German-style buffet (along with some American favorites), you will be entertained by costumed musicians playing traditional German music. Dancing is encouraged, and on our two last trips, our two young boys, a preschooler and a toddler, had a blast dancing with other kids on the restaurant's dance floor.
Other fun favorites include:
While slightly more pricey, you may want to also take a look at 'Ohana (for dinner), a Polynesian family-style restaurant, and Coral Reef, a wonderful and fun experience for kids (and adults) where you can dine in front of stunning underwater views.
Enjoy your vacation.