I really appreciate everyone's help and imput here!
Good to know that some have had success in this battle. It's a shame these plants are so unruly because they are actually one of my favorite berries. I'll probably keep a small amount at the very back of the property as a barrier and be able to enjoy some berries as well. Great idea about having it along the road Daron Williams. Good job in your battle with them!
I'll keep everyone posted about my success or lack there of. I was thinking along the same lines Kurt Rider and Kim Goodwin, as the ground gets softer under the cardboard/chips then the roots will pull up more easily.
Also Kurt, we have entertained the idea of a pig doing this work for us... I just have a family member whose not won over to this idea yet.

But it's by far my favorite idea!
I did hear of someone eradicating the blackberries by planting grass seed and the sod choked out a lot of new growth. He cut down the new canes with the lawn mower as he mowed the grass. Eventually had none.
But that takes years and we are itching to grow our own food and sell surplus at the farmers market and can't wait years to begin.
Thanks again and any other ideas appreciated!