Don safety goggles and thick gardening gloves for this project to protect yourself from shrapnel. Work in the summer, when the plant is fully grown and you will not disturb birds' nests.
Start from the top of the vine area and pull outwards to dislodge the vine from the wall, tree or arbor to which it clings. If the vine is not too old or established, you should be able to pull away most of the vine.
Cut away any areas that cling to the structure with sharp pruning shears or loppers. Take care not to cut into the bark of the tree, if the vines are hanging on a tree.
Make as many cuts as necessary to dislodge the vines.
Pull ground vines up by the roots. Remove all roots with a shovel or spade, or till the ground thoroughly when finished to break up any roots. Any remaining roots have the potential to sprout new vines.
Dispose of all pieces of vines safely in a compost pile or trash can. Any pieces of vine that remain in the soil, or are dropped or buried by accident during disposal, have the potential to create full new vines.
Cut and pull the vines off each year you see them growing. It may take two to three seasons to completely remove all traces of the vines from the area.