I don't believe in herbicides, they kill far more life than those intended. So, confronted with a backyard threatened to be comsumed, and deciding to limit the destruction. I bought and borrowed some good cutting instruments, both long reaching and shorter, sharp shovel, heavy duty gloves, old tough jeans & high boots. These last items limit the damage to you, as the bushes want a taste of you (defense on the plants part).
Taking a small area at a time, I cut as much as I could starting at the top, then cutting smaller sections, until the bottom. Then using the shovel, digging them up with more cutting as necessary. If reversing (bottom to top) is easier on you, do it. All the cut material can be mulched at the site or in a compost pile. Shred into small bits if possible.
You might contact a local nursery or plant group to see if anyone wants the roots or rooting plants. You might even make money! This is probably not a one weekend project.
The benefits are: You are rid of blackberries. You are being in touch with your yard and nature. You know you are keeping the world clean of poisons (herbicides do some very nasty things to other living beings after it leaves your property). It is excellent exercise and personal stress release.
Cursing in low tones recommended unless alone, though ocassional yells can bring help! Please keep in mind, the Good points about these plants. There are many. Good luck!