i’m doing the cut and pull/dig-up-by-the-root method. it’s tedious. buy leather gloves. and a bunch of 32-gallon cans.
then, if the soil conditions are right, plant native wild strawberries. i’ve read that they compete with blackberries pretty well. or maybe douglas spirea or swamp rose.
http://www.soundnativeplants.com/catalogterms.htm
this site has excellent information for native plants to replace the blackberries, with information about root systems, soil conditions, light conditions, etc. separated into tress/shrubs, grasses, and groundcovers. has links to pictures and tons more info at the UW horticulture website.
good luck!