From U of F:

Weed Control
It is extremely difficult to gain control of weed problems in a blackberry planting. Methods to control weeds in blackberry plants include mulching, cultivation, herbicides, and mowing. The yield results of a mulching trial conducted at the UF NFREC��"Monticello were discussed above. The plastic mulch treatment was the most effective in controlling weeds, followed by pine bark. Mushroom compost and bare soil treatments were equally prone to weed problems. Cultivation, whether it be by hoeing or disking, should be very shallow and no more than 2 inches in depth. Blackberry roots are very shallow and easily injured by deeper cultivation. Contact your county Extension agent concerning specific herbicides that can be applied to control weeds in blackberry plants. It is beneficial to mow grass in between blackberry rows to enhance air circulation and minimize the seeding of in-row strips.

Here is a link that might be useful: http://edis.ifas.ufl.edu/hs104

Mike,

You can try using a stiff wire hook (old style wire coat hanger) to pull the large tall weeds away from the blackberry canes so you are able to grab onto them to yank them up. It's kind of slow going but it works and saves a heck of a lot of ouches when weeding my wild blackberry patch.

Lou

I use a rake to hold them up while my husband gets the weeds out

Maybe some of those thick leather gloves that go up to the elbow that are sold for working around rose bushes would help.