Hopefully Mr. Frazier (the reporter) checks this thread more than once to see the new responses! 
Here are a few places for "Relaxing Time at WDW" that I've added to our trip information. (we're going for our honeymoon in August - first visit to Disney!) Sorry if some of these are repeated from above, I just cut & paste. Most of these suggestions are from AllEars and the DIS, particularly the WDW Secrets thread.
Animal Kingdom
Take the Discovery Island Trails, where you'll find several peaceful benches where you can sit and rest, listen to the music.
Go to FlameTree BBQ for lunch (or just order a beer in the afternoon), then go to the left of the building and find a nice table to sit at. They are all in the shade, and there are quite a few down by the water, very quiet.
Sit by the exit of Kali River Rapids (before you go over the bridge toward the ride) on a bench behind one of the kiosks, and watch the monkeys.
The train ride to conservation station is relaxing.
Across from the Dawa Bar on Discovery Island is what appears to be an old fort. Don't assume that the area is off limits -- there are actually tables and chairs in there! It's the perfect quiet spot to take a cool drink or a snack and get out of the flow of traffic.
EPCOT
Japan - The Matsu-No-Ma Lounge in the Japan pavilion is a quiet retreat from the hub-bub of a busy theme park day. Walk up the steps of the pavilion's department store to find the unusually peaceful lounge on the second floor. There you can sip sake (or another beverage of your choice) while gazing out of the large picture windows, looking out over the lagoon, watching the crowds go by or the preparations of that night's showing of IllumiNations.
Japan - Sit in the gardens & look at the Koi fish.
Norway - Get a drink. Sit on a shady bench & eat an open-faced sandwich & share a desert from the Norwegan Bakery....along with a couple more beverages. People watching & relaxation at its finest!
Mexico - There is a walkway to the left as you exit Future World and head towards Mexico. There are some benches along the way. We spent about an hour there relaxing and having some popcorn while the kids watched the birds on the lake. It was perfect just to relax and soak up the atmosphere.
The path leading to Test Track has a beautiful Butterfly Garden that features plantings designed to entice the winged beauties. It's a quiet spot not too far from Cool Wash misting station on the way toward World Showcase.
MGM
Sit outside of Pizza Planet in the shade near the Miss Piggy fountain.
Magic Kingdom
Main Street Exposition Center, Sponsored by Kodak (near Tonys) - Go into the building right after you come out of the railroad station. It has a little movie theater that shows classic Disney cartoons. Since it is somewhat dim in there, you can have a short "cat nap." Go back outside and sit at the top of the train station. You go up the steps and right to the center of the balcony. A perfect birds eye view to sit and look straight at Cinderella's Castle.
Need a break from the busy Main Street bustle? Pop in for a soda and a hot dog at Casey's Corner, on your left as you're walking toward the castle, between Main Street and the Crystal Palace Restaurant. Tucked in a back room are a few quiet tables and benches, in front of a wide-screen television playing old black-and-white sports-themed Disney cartoons! Or, if you venture to the outside seating area, you might stumble across the ragtime piano player who performs periodically throughout the day. He takes requests!
Take the train around MK. It provides a different prospective and a nice break too.
On the side of the Hall of Presidents' building, across from the Liberty Bell in Liberty Square, you'll spy two old-fashioned rocking chairs on a small brick porch. They're not just props -- you really can rock away a few minutes (or hours) from this comfy spot. Stop at Sleepy Hollow for some caramel corn or an ice cream sandwich, and bring your treats back to the chairs for savoring as you people-watch from this out-of-the-way spot.
Behind the Ye Olde Christmas Shoppe in Liberty Square, there are lots of benches in the shade.
The "Story time with Belle" gardens in Fantasyland are nice, if you can get there just after a show. There is a lot of shade and benches, and it is tucked neatly away, so most people just walk right on by and don't even notice it!
There is a walkway between Tomorrowland and Toon Town at that is wonderful. It has benches to sit on and you can sit and relax and watch the train pass every few minutes with happy families waving to you from onboard. Not many people use this walkway and for Magic Kingdom, it really is relatively quiet.
The Carousel of Progress is relaxing ride, but not boring. (Tomorrowland)
Also relaxing is the Tomorrowland Transit Authority. You can just keep riding this ride without getting off if you want. It's a nice place to rest your feet. (This is a way to see the inside of Space Mountain without riding it)
Near Space Mountain (Tomorrowland), take a stroll around the side of the building along the shady path. You'll come upon a circle of private benches that afford you an excellent view of the WDW Railroad as it circles the MK.
Around twilight, take a walk through the All American Rose Garden outside Cinderella Castle (between the Castle and Tomorrowland), down near the old Swan Boat dock. It's off the beaten path, but offers beautiful views of the castle and the heavenly smell of roses surrounds you. Occasionally, you'll even find a family of ducks, with their ducklings, waddling amongst the blooms.
Leave the park for a little while at mid-day. If you want to do something quick, you can hop on the monorail (resort side) and visit one of the monorail resorts, or just enjoy going around the loop. If you have a little more time, you can walk to the left of the monorail and hop on one of the resort boats. Take a ride around the lake and stop to visit a resort for drinks or lunch. For a longer break, take the boat over to Fort Wilderness. They have a petting zoo and a great spot for a buffet lunch. You will feel like you've traveled back in time and that you're nowhere near Disney.
