In the Magic Kingdom, there is (or at least used to be, haven't been in a few years,) a quiet area over near the old Skyway station in Tomorrowland, with rest rooms and shade and benches.
At Epcot, the area by the river in France is a nice quiet area. (Or at least, it used to be, there going toward the water between the perfume shops. Not sure if construction has taken over there.)
At the studios, the area near the Indiana Jones shop near PrimeTime feels somewheat remote.
At the Animal Kingdom, the path from Pandora to Africa is a good one.