Blackberries are virtually impossible to remove completely even with spraying and or digging the roots, because the birds keep planting them. ;o)
My understanding is that if they are kept mowed down, they will 'eventually' starve and die back, but you'll probably be fighting them for a long time.

A helpful person at a local nursery advised that we use a paint brush to "paint" round up onto the freshly cut portion of the blackberry stem once we'd removed the majority of them and cut them way down. She said this would limit the nasty chemical from spreading to the surrounding environment. Not sure how effective it was - just did it a few weeks ago, then put a fence up in front of the area. I did find a natural product on the web, out of Seattle, called "Blackberry & Brush Blocker" which was a vinegar based spray:
http://www.seattlepi.com/lifestyle/homegarden/article/From-the-Garden-New-vinegar-based-sprays-safely-1056568.php
but I couldn't find anywhere to purchase it, even after checking manufacturer website, Amazon, etc. Good luck!