Cut the vines using pruning shears or a machete. Work your way inside the vines to the root level as you cut them. You want to cut through and disable the massive vine structure for easier removal. Cut the vines from any trees they have grown on. Cut only the vine and avoid disturbing the tree bark.
Pull the vines from the woods and lay them on open ground in direct sunlight. The sun helps dry out and kill the vines.
Dig the vine roots from the soil with a shovel, or use a tiller to grind them to the top of the soil. Removing the roots keeps the vine from redeveloping. Lay the roots in the sun with the removed vines.
Place the dried vines and roots into trash cans or trash bags for disposal, or call your local waste management center and ask about the proper way of packaging for yard waste removal in your area. If you have a compost bin, place the vines and roots inside or place through a chipper to reduce their size.