Pull them out and dig them up, getting as much root out as possible, over and over again, til they give up. Sorry there isn't an easy answer. I've found that the biggest brambles can usually be conquered in about three passes. The roots seem to get shorter when they have to make new top growth and a huge "crown" of roots becomes manageable after all the individual bits have been hunted down over a period of months.

Ivy is physically easier to get out as it mostly stays at or near the surface. I just crawl about tracing all the bits and pulling them into a pile to take away and burn.