Thanks to current technology, running a dishwasher actually uses less energy than washing dishes by hand. According to a recent Ohio State University study, energy-efficient models use about six fewer gallons of water per load than you would use washing the same number of dishes by hand. Using six fewer gallons of water means heating six fewer gallons of water thus saving energy.
Another study at the University of Bonn in Germany has proven that an efficient dishwasher uses only half the energy, one-sixth of the water and less soap than hand washing. This study also rated the cleanliness achieved and found in favor of the machine verses washing by hand.
Good question, thanks for asking!