Will, I bought a tooth bar for my bucket this winter. I have had some good success by running the bucket into a group of vines, curling the bucket slowly down and backing slowly away. I try not to do it on tall trees with dead limbs, and I wear a hard had. This works especially good with multiflora that has grown up into the trees. When I get a bunch on the ground, I tip the bucket all the way down and float it as I back away from the tree. The tooth bar "rakes" the stuff to where I can push it into a pile to rot down or otherwise dispose of.
On the real big vines, I use a log chain (that's what I have) and the draw bar, just like Bird does with his cable. The only thing I notice is the vine seems more apt to break, rather than pull loose, during the winter and after having cut them previously.