The safest method is continuously cutting/pulling up the plants on a weekly basis. Doing this frequently will weaken the plant.  You can also dig/pull up the roots but be careful about leaving pieces of root as they can produce a new plant.  You will still need to be vigilant because birds love blackberries and drop seeds (with a bit of fertilizer.)  I usually find at least one or two small blackberry plants in my yard every year.

If you are open to using chemicals, in late August let the vines go (it may be too late for them to produce mature fruit) for a couple of weeks and then spray the foliage with Crossbow (this is a trade name) http://www.cdms.net/ldat/ld02h006.pdf  It is more effective on woody shrubs than glyphosate (e.g Roundup)  It is a non-selective herbicide so you will need to protect any plants.  Please be sure to read the directions on the label twice and follow all safety precautions when applying.  I usually don't recommend using this method in the spring because the application is more effective when it's hot and dry in late summer.