Picnics in the Parks?

Unfortunately, grassy areas are usually railing-ed off, so quiet areas have to be benches or some such in the parks.

In Animal Kingdom, there are areas off of the path between Asia and Africa, down by the water, that have tables and chairs to rest/eat. Usually these are pretty abandoned because everyone's rushing by to get to rides.

Also in Animal Kingdom, you can go to the little courtyard in Harambe Village. it's across the "street" from Tusker House restaurant and next to the path to Asia. This is where King Louie and Baloo can often be found, but nobody's ever back there.

In Epcot, there's a sculpture garden near Imagination that's usually pretty empty.

(A lot of these places are away from CS restaurants--I wonder if they'll put your food in a bag for you.)

CS restaurants aren't really crowded outside of main lunch times, so you'll probably be fine if you visit around 11am or after 2pm.

Another quiet spot is the rocking chairs at MK. Over by Liberty Tree Tavern and up on a couple of the "porches" of the "houses" in Liberty Square you can find some rocking chairs. Also by Hall of Presidents.

The "working remotely from WDW" article on this site has a few quite spots noted, as well. Maybe not all of them for eating, but at least for "getting away from people" for a while:

Specific Park Locations:

[OL]
[LI]Magic Kingdom: The back of the Town Square Exposition Hall: Walk past Tony's Town Square podium and the little gift shop. You'll see a long, wide hallway leading toward the back of the building. There are a couple of large rooms past that hallway with picture spots and a continuous movie. This place always seems dead and has few visitors. If it's crowded, move as far down the hallway as possible and you should find some quiet space.
[LI]Epcot: American Pavilion's American Heritage Gallery, Morocco Pavilion�s Gallery of Arts and History, Norway Pavilion's Stave Church: All of these places are quiet and actually relatively unpopulated simply because few guests know they even exist! The Stave Church likely gets the most traffic of the three. All are indoors and may or may not have background music.
[LI]Epcot: Living Seas Pavilion: The Living Seas Pavilion has seen more traffic since the addition of Turtle Talk with Crush, but there are still several areas of the pavilion that are quiet and set apart from Epcot's crowds.
[LI]Epcot: Odyssey Center: The Odyssey Center is on the bridge between Test Track and World Showcase (closest to Mexico). Disney now holds special events in this building, which also houses a baby care center and some of the most deserted restrooms in Walt Disney World! Guests can wander in and out of the Odyssey Center when it's not in use, it's completely silent inside, and it makes the perfect spot to take a scheduled conference call. I've done this several times and have never had a problem.
[LI]Hollywood Studios: The Magic of Disney Animation: You can bypass the animation tutorial and head right into the exit area where there are exhibits about animation and the history of Disney and Pixar movies. This place is usually pretty empty, is indoors, and has comfortable padded benches. It's also large, so should a group of guests exit the tutorial and raise the noise level, you can venture to another part of the building to continue your work. [/LI][/OL]

Other nice places to go for a meal (especially counter service) are the resorts. That way you can check out resorts you've never been to, and because everyone's in the parks mid-day, you can hang out in relatively empty surroundings while exploring and having a truly quiet and enjoyable lunch.

Just came across the quiet spots in Disney World thread. Might or might not be helpful...