There is only one restaurant at Downtown Disney that is actually operated by Disney (Cap'n Jack's, which is not particularly Disney themed. Just to note, it's been named Cap'n Jack's for many years and the name has nothing to do with Captain Jack Sparrow.) So there really are not any Disney-themed restaurants there.
If you are looking for Disney-style animatronic type theming (without them actually having anything to do with Disney) go for the Rainforest Cafe (jungle animals) or T-Rex (dinosaurs).
If you want a restaurant themed to a certain culture you can try Raglan Road or Bongos. But you are not going to see any Disney characters or Disney-related situational theming in any of these places.