Using Roundup in the fall when the vine is weak is a good time and works well. Seeds can live up to 50 years, so get rid of any you see. Removing plant and roots about 6" deep should work, too, but that's a lot of work.

A few non-poisonous solutions:

-Full strength vinegar that you can make yourself with a few bottles of cheap wine OR you can purchase Surig Essig Essenz Concentrated Vinegar (available at GermanDeli.com) in a pinch.

-Boiled water OR pure lemon juice poured on plant (boiled water works well on fire ant nests).

-Crowding out the plants with other plants, such as oats or other tall grasses.

-Sawdust from black walnut trees (toxic to plants).

What won't work:

Smothering in plastic - especially in windy areas.

Tilling the soil - this just germinates dormant seeds and introduces more plants through cut roots and stems.

Hope these ideas help!