This may seem like an odd answer, but I always find Sci-Fi Dine-In at DHS to be the most relaxing restaurant at the parks.

Yes, there's a giant screen playing B-movie clips relentlessly, but because of this, everyone's glued to the screen and the place is often so quiet you only hear the occasional clank of silverware. Plus it's very dark, and the A/C is always at full blast. I always want to take a nap in our car after eating!
If you don't want to venture into a park, many resort restaurants are often quieter. Definitely not the food court at All-Stars, but many others. Kona Cafe at the Poly is one of our favorites, food wise, and it's generally pretty quiet, 'Ohanas draws the crowds and noise. The Wave at the Contemporary has a calm atmosphere...ditto Sanaa at Kidani Village, Turf Club at Saratoga Springs, and Mayan Grill at Coronado Springs.
CS places tend to be noisier in general it seems, but Columbia Harbor House at MK is usually pretty relaxed, especially the upstairs area! In Epcot, Tangerine Cafe and Lotus Blossom can be quiet because they don't draw huge crowds, but their seating is pretty limited.
...I probably should have just asked you where you were willing to travel to, since there are so many different potential answers based on your plans for the day and whether you're going to a park or not, but anyway it really isn't as hard as it would seem to have a relaxing dinner at WDW!
