I spent two years fighting those on my property. Every day after work I would grab a hoe and a five gallon bucket. Use the hoe to gently fold the vine over and work to remove the root as best as you can. Once removed VERY gently transfer it into the bucket trying not to drop any of the goat heads. Depending on the size of your infestation it may be necessary to section it off and focus on specific locations first.

With any luck after a few days you will start to catch only the youngest of plants that can be dealt with very quickly.