Round-up should do the trick.

Your invasive hell-plant is my protected baby. I have 2 blackberries and go to great lengths to keep them alive through the -35F temps we get each winter. To each their own :)

I second don555's sentiment... I feel this way about most plants that folks from other places rant about (mint included).

Then again, the line between invasive and useful is especially thin here in NM. Most of the plants that are invasive elsewhere are especially welcomed here because of their vigor.

Digging up the roots should have worked... brambles are extremely small and shallow-rooted, so you must have just missed one or two plants.

I've always just pulled the suckers out. They will try to come back several times but eventually they stop and come back else where. It's no different than pulling weeds every few weeks. They pull much easier when you first notice them as a small sprout. I think controlling grass in a raised bed is much worse than brambles but that's just me.

Maybe I just have to be more aggressive and pull them out when they just peak out of the ground. Thanks!