Apparently, dishwashers beat usual methods of handwashing even when the environmental production cost of the dishwasher is taken into account (though that estimate doesn't seem to include the cost of transporting the appliance, mind you).

The same study suggests one method of manual washing that beats dishwashers: the two basins methods (you fill one sink, plugged, with hot water, scrub the dishes inside that sink without running water, move the scrubbed dishes to a second plugged sink filled with cold water to rinse them).

Finally, that study bases its estimates on the U.S. national average electrical grid carbon intensity, and assumes water is heated in the home through natural gas; and most of the energy consumed is for water heating. So if you're living in, say, France (where 90% of your energy comes from low-carbon sources such as nuclear or hydro-electric dams), and your water heater is electric, the results might well be different.