Elizabeth---HI
Those sound like Air Potato vines. In order to kill them you have to dig up the "potato". You know those brown things you see on the ground? They eventually take root (like a flower bulb) and grow into those vines. They are a mess to get rid of, I keep digging them up and throwing the potatoes away, but they seem to love my fence and trees,
The reason they are such a problem here in FL is that they kill the tree over time.
Here is an article about them
http://www.sptimes.com/2002/02/24/So...reduced_.shtml
http://aquat1.ifas.ufl.edu/diobul.html
HTH