Calling all green thumbs! How do I get rid of unwanted, wild raspberry briars (which are thorny) without destroying my flower bed? I have hasta and some kind of mountain laurel (or shasta daisy in a bush formation) growing in a flowerbed I did not start. It came with the house. Later, the wild raspberries moved in, and they’re taking over! How can I get at the root of this problem? 
I wonder if they will grow here…? (South Texas)
Hey, you can come here and try to transplant 'em! You might have something there. There’s a local, online board where people post all kinds of free things. We once posted our rose bushes (because they were tearing up our screened-in porch due to poor planning by the original owner of my house), and someone came and took 'em! Seriously!!! 
(I couldn’t do the labor myself due to a bad back.)
If they’re out of control, you’ll probably need someone to dig them out for you.
Here in Florida, a trick we use sometimes is sprinkling some baking soda on the foliage of the unwanted plant/weed. Make sure you only get it on what you want to die. It has to stay dry for as long as possible. Don’t overdo it, or nothing will grow there.
The best time to deal with them is late summer, but spring will help too. Get a concentrated form of Roundup (I think 41% is best, but 20-30% will do it). Cut the stems of the plant, and dab the Roundup on the stumps. If you do it in late summer, just before the leaves turn color and drop off, the chemical goes into the roots much more effectively, but doing it several times in summer can work, too.
I often put a bit of dye in the Roundup, so I can tell what I’ve treated.
NOTE: This will kill the entire plant, so if you wanted to keep some blackberries alive, you are out of luck. But, keeping any blackberries alive assures that you will have them in your garden again!
Either use a weedkiller gel that you can paint directly onto the offending plants (as Fetchund suggests), or if you don’t want to use chemicals, just keep on removing all of the stems and shoots you find above ground.
In order for that to work, you do need to have access to the entire top growth of the plant (if it’s creeping in under a fence or growing through a bush, you won’t be able to remove it by repeated pulling.
It takes a lot of patience and diligence, but if you keep on removing the top growth of a plant, it will eventually die, because it can’t manufacture food (exceptions for parasitic plants, etc). I rid myself of a patch of mare’s tails this way - and they’re supposed to be quite a persistent weed.