One of my favorite hideaways in MK is the little side street on the right off of Main Street - I think it's called Center Street. There are some cute little café benches and flower pots at the back and there's rarely anyone there. It's not completely quiet, as there is some ambiance noise - mainly the "music lessons" heard through an open window up above - but it's cute and not too loud or distracting. There are even a couple of pay phones hidden back there, so this area is designed to be a quiet spot.
There's also a wonderful rest area set up in the tent farthest back in Storybook Circus, to the right of where you meet Minnie/Daisy/Donald/Goofy. Chip and Dale were meeting there for a while, but I'm pretty sure they've moved back to Fronteirland now. This tent is set up with lots of comfy arm chairs and small sofas, is completely shaded and very quiet. There are some outlets for charging phones, too.
I like Tom Sawyer's island, too, but it's a bit of a pain to get back there sometimes since you have to take the little rafts.
In Epcot, I recently discovered the little patio area down below the bridge from UK to France. It's not shaded, and the only seating is on the planters, but it's a pretty spot at the edge of the lake with a great view, and it is separated enough from the bridge that it's pretty quiet down there.
In Japan, there's a pretty garden area up above the little koi fish pond. No seating, but a very pretty spot.
The rotunda area in the American Adventure building is beautiful and very quiet in between the Voices of Liberty's performances - there are some padded benches in there, the paintings in there are actually pretty cool when you take the time to look at them, and the American Heritage Gallery is well worth checking out. And, of course, don't miss the Voices of Liberty performances themselves!
Hollywood Studios doesn't really have much. There's a spot in the Chinese Theatre courtyard off to the left (opposite from where the Great Movie Ride line is) that doesn't get *too* much traffic, and you can sort of sit on the curb around the little rock/plant feature there. The Star Wars characters come out from a door near there before their stage show, though, so it just depends on timing. It can be nice to take a bit of time to really look at the handprints/signatures in the courtyard, too - there are some pretty cool ones. (My favorite is Judy Garland's signature with an imprint of the ruby slippers!) There are also a few benches and a short retaining wall you can sit on around Echo Lake near Gertie the Dinosaur's ice cream stand. Again, not really quiet, but less traffic than other areas in the park.
In Animal Kingdom, I love both of the main animal trails - Pangani and Maharajah - plus the little nature trails back behind the Tree of Life. There are a few other little animal exhibit spots that are nice to just rest near - I especially love the otter near Pizzafari when he's visible, but you're right near the main path, so it's not particularly quiet there.