They make the most wonderful jellies. Think of them as blackberries, with a disposable core and seeds that should also be removed.
The tree we got ours from was a Persian Mulberry with a huge berry, marvelous for eating fresh or jellies. Putting down sheets of plastic and shaking the tree would do it nicely. With old sheets...well, everyone knows how permanent berry stains are, but an old one from a thrift store, washed first is the way to go.