My personal favorites are:
Carnation Cafe in DL - great location for people watching along Main Street. I prefer it not so much for breakfast but for lunch-dinner-dessert only meals (good meat loaf stack and yummy Matterhorn sundae & apple pie a la mode). Seating is limited so definitely make a reservation for this, as tables fill up fast and the line of people waiting can be long.
Blue Ribbon Bakery in DL (counter service)- Yummy, ooey, gooey, messy cinnamon rolls, blended mochas and other baked treats.
River Belle Terrace in DL (counter service) has a nice location at the intersection of New Orleans Square and Adventureland, and serves the famous Mickey Mouse pancakes for breakfast, as well as some fairly thick turkey sandwiches for lunch and dinner.
Plaza Inn in DL is counter service, "buffeteria-style" food and the interior of the restaurant is quite beautiful. The portions are quite lare, and the chicken is very popular here.
Bengal BBQ (counter service) in DL serves various skewers (I am partial to the chicken skewer), which are enough for a light meal.
Wine Country Trattoria in DCA has a nice ambiance (try to get an outdoor table). Yummy meatball sandwiches. Other people are partial to the lasagna and my friend enjoyed the Caprese sandwich and the huge pizzetta salad. They also served an eye-catching blood orange sorbet.
Taste Pilots Grill in DCA (counter service) serves decent burgers and shakes and all the expected accompaniments, and it has, in my opinion, the best soundtrack. While you eat your burgers, you will be treated to various space travel/aviation-related songs such as "Space Oddity" by David Bowie, "Rocket Man" by Elton John, "Fly Like an Eagle" by Steve Miller, etc., etc. Seeing that I grew up in that kind of 70's classic rock age, I would go there just to listen to the music!!
Storytellers Cafe in the GCH is a really nice, cozy, warm, earthy kind of place with a fireplace and rustic sort of atmosphere. I prefer it for the non-character meals. Lunch is good there and they serve the best spaghetti and garlic bread for dinner!
Goofy's Kitchen in the DLH is my favorite of all the DLR restaurants - it happens to be a good restaurant with a wide variety of food choices...that also just happens to have characters!
As for the non-Disney specific restaurants that can be found in Downtown Disney, I like Rainforest Cafe for breakfast, lunch or dinner, Naples Ristorante (try to get an outdoor table to soak in the fun atmosphere of DTD), and I like ESPN Zone (good portion sizes, good food - very busy, noisy atmosphere, however).
I have not tried them yet, but I hear great things about Cafe Orleans in DL (pretty consistently), Ralph Brennan's Jazz Kitchen in DTD and Napa Rose in the GCH.
Oh, and be sure to not leave DLR without grabbing a delicious, sweet, juicy, refreshing pineapple spear from one of the outdoor vendors on Main Street or in DCA. Those spears are SO delectable and addictive - DLR really does fruit well, as most of the melon, berries and pineapple I have had there is out of this world. (So is the bacon, as a rule!)