The pruning of Blackberries is simple but you must understand what type of growth habit you have.  Some will fruit on first year wood and some on second.  

In general all you need to do is remove the dead canes and leave the ones still alive.  The canes which grew last summer and still often have a few leaves will be the fruiting wood for this season.  There will be canes with disease or damage from bending which you will want to remove.  

Take any diseases canes out of the area.  Put in dumpster and get off your land.  Bruning can be alright but it is wise to simply let them go with the trash.

Information Link - The Ohio State University