The easiest way I know is to mow them. I don’t know whether that is practical for you. If you turn that area into lawn the problem is solved. You could even mow just a time or two a season if you wanted and have a mower that will handle that.
Keeping trimming them back before they get any serious leaf. As they keep returning, they’re using energy in their roots that’s not being replenished by the presense of leaves. After a while, they’ll weaken and die. Be dilligent and patient – it can take a while. They grow extensive root systems, so make sure you get all of them in your yard or an overlooked plant several meters away could replenish the energy the root needs to survive. Same goes for those darned green-briars. Digging or poisoning just doesn’t guarantee removal – but patience and consistent clipping will.
The blackberry question is a good one. At our last home (2 ½ acres) the blackberries grew almost to the house – we chopped them down and mowed them whenever sprouts came up. We did keep some at the back of the property for jam. They are almost the Pacific NW version of kudzu – so fast they grow. They are also very prickly. Now that we have more room we have a »blackberry patch’ on one side of our property. We are able to keep it under control only by mowing weekly during the growing season.
The kind we have a problem with here in the Pacific Northwest are the Himalayan variety. These non-natives are very invasive and choke out native species of huckleberry, salmonberry and even our native trailing blackberry. They grow so rampantly, they can quickly engulf a small outbuilding in one season. Oh, and the berries are insipid, watery and not nearly as flavorful as other varieties (esp. Out native one).