I think you mean brambles, rather than blackberries. The former are a sign of a neglected garden, the latter, usually with few thorns, are the domestic varieties, which aren't a patch on the wild ones. As with most unwanted plants, there is no substitute for hard labour. Most things can be dug out and will eventually give up the ghost if you are persistent. If you cut the top growth back to near ground level and then dig out the roots, any re-growth is quite easy to deal with.