Trumpet Vine Problem
 

We have remnants of trumpet vine running rampant in our gardens/yard.
Anyone have good suggestions for eliminating this stuff?
I've been using Round-Up on the little ones we find, but it actually takes
several applications to kill it - and we're not convinced the brown ones are
dead-dead, but they're still very difficult to pull out of the ground.
This stuff is EVERYWHERE and it's impossible to just pull it out......
(We also aren't big chemical fans, but white vinegar had NO effect, so we
thought we'd try the Round-Up.)

Anyone?

Thanks,
Nicole