I have a 10 year old asparagus patch that's needs to be moved. What do I have to do to transplant the roots?
Those roots will be huge and very difficult to dig. You didn't mention how large the patch is or how many plants you want to move, but each one can probably be divided into several smaller plants. Dividing the mature roots will rejuvenate them and should increase production. It may take a year or two before they really start to produce though.
Dig completely around each clump and then start digging toward the center of the clump. The roots will probably extend out for several feet from the edge of the clump and at least 18" down. After you lift out the clump you can try to separate the plants with your hands, but the roots will be hopelessly tangled the closer you get to the center of the clump. Normally it's best to prepare your new bed ahead of time and get your plants into the soil as soon as possible after digging them. But I moved my asperagus a couple years ago and left some of the roots out inadvertently for several weeks. When I finally got them in the ground, they took off and grew just fine.
Do you have to move them? They're beasts. I offered to dig up a few 3 year old roots for a friend and it nearly killed me. Massive tenacious crowns and roots, well described by Cheryl.
Yes, the chore of the crowded asparagus patch. Best to have teenagers around for help on that task. The only thing to add to Chery's apt description is that you'll be happier with newly-sharpened shovels/tools and a couple good strong forks. You won't save all the roots so pick some you'll save and don't feel bad about the rest.
Oh, and let some folks have some asparagus and offer to give them some from the move, and come by for a few easy minutes in two Saturdays to help get a couple! Easy as pie!