I grew up in Woodinville and spent much of my childhood clearing blackberry bushes.
What I found to be the most effective was a gas-powered weed trimmer with a round metal saw-blade attachment (they sell them specifically for blackberries and other thick-stalked weeds) -- take everything down to the ground, and then dig out as much of the root as possible, and then use a ground tiller to tear up the ground and rip out any more roots, and then carefully comb all those roots out of the soft ground, and keep an eye on the area and till it again once you see any new shoots.
We tried a flame thrower and actually caught an entire underground root system on fire -- have you ever seen an underground fire? really weird! and dangerous.