Catching up on dishes: The Dishwasher Edition.
1. Load the dishwasher up with the dishes that are the easiest to prep - non-crusty dishes, mixing bowls, etc. The key for round one is to get the dishwasher full and running as soon as possible. Don't worry too much about conservation of space in this round, you just want to get a bunch of stuff out of the way. Run the dishwasher. If you have the choice, make sure to use the hottest possible setting for the final rinse.
2. While the dishwasher is running, prep the dishes for round two. Depending on your dishwasher, this could mean anything from just scraping the big chunks off into the trash to soaking, scrubbing, and practically washing every dish (and wondering what the point of the dishwasher is, anyhow). Stack the prepped dishes in piles of similar items to make loading the dishwasher for round two easier.
3. When the dishwasher finishes, open the door immediately. Pull out the racks and shake them vigorously to get as much water as possible off of them. (Do the top rack first, otherwise you will just shake the water from the top rack back onto the bottom rack.) If you do this while the dishes are still hot from the rinse cycle, they will dry quickly (and almost completely spot-free).
*Edited to add: You don't need to pull the racks all the way out, just shake them back and forth on their tracks and leave them sticking out as much as possible. The more they stick out, the quicker they will dry.
4. Go do something else for a few minutes.
5. Put away the dishes. Most of them will be dry enough to put away after a few minutes of drying, assuming you followed step 3 correctly. Plastic things might still be wet because they don't hold the heat enough to dry themselves, so just move any wet items to the dish drainer to finish drying.
6. Load up the dishwasher for round two. With your pre-sorted stacks, this step shouldn't take too long. Use your Tetris skills this time, to fit as much as possible into the dishwasher. Your pre-sorted stacks will come in handy for spacial organization, too.
7. While the dishwasher is running round two, assess the remaining dish situation. If it's enough to warrant a third dishwasher round, return to step 2. If not, just wash the remaining few dishes by hand.