Yes they do, whether it's remembering the death of their owner or another dog or cat that they were friends with. They grieve much like people do just like people each dog is an individual and will grieve in a different way. You should read the book 'Greyfriars Bobby' and many stories about dogs that grieve. I've seen dogs leave the home and walk miles to a cemetery to lay across their owner's grave. Just like people, in time the dog will come to terms with it with the help of the family or its new owner.