Smilax, aka Catbrier, Greenbrier, Prickly-ivy, Sarsaparilla, Sarsparilla and Smilax is often considered difficult to remove once growing because most people know not how it grows and is spread, a disconnect from nature. The rhizomes in the soil spread out and about and are deep enough that a fire will not harm them and this plant is often one of the first to recover from that. Birds also eat the ripe berries and drop seeds of this plant whereever they fly.
The best control, and the easiest is to keep the plant cut down so no green growth has time to manufacture nutrients for the rhizomes to store that will promote that to grow some more. If the green growth is kept cut the rhizomes will soon die out and so will the plants, except for the new ones birds plant.

Thanks! So I eliminate the vines until they stop growing from the rhizome. That should be easier than digging them like the extension agent recommended. We've dug up some rhizomes the size of my foot. I might cut the vines from one half of the yard and round-up those on the other. Same m.o., maybe one will work better.

At our house, we dig up the roots. We find this is the quickest, most effective method. Be sure to also dig up any potato-like tubers connected to the root.