Thornless Blackberry http://www.durgan.org/URL/?CMASD 8 September 2009 Thornless Blackberry. There are two plants on the 8 foot trellis. These two plants were purchased two years ago from a supplier and there was no name, except thornless blackberry. The berries are quite sweet and are excellent, provided they are picked when just about ready to fall off the plant. This is my first experience with blackberries in the home garden. This is the first picking of the fruit.