Take two aspirin and follow a "desensitization" program. No matter which appliance you tackle first, the idea is the same. Just take it one noisy diversion at a time.
Begin with a dog treat party and play with your pooch. A few minutes into the game, have someone briefly switch on the vacuum cleaner in a far-off room. Continue playing as if nothing is happening. If your dog doesn't hesitate, keep going. Bit by bit, have someone steer the vacuum closer and closer. At the point your dog becomes distracted by the monster vacuum, you've gone too far.
With an appliance like the dishwasher, which can't easily be steered around the house, direct the dog closer and closer until it becomes bothered. Reward the dog with praise and treats for simply having a good time and ignoring the dreaded dishwasher.
Some behaviorists recommend you start the desensitization program by tape recording the offending appliance(s). Start by playing the tape at a volume so soft your dog can barely hear it. Play the tape while your dog eats so it will associate the sound with its beloved dinner. Slowly turn up the volume. If your dog balks or appears upset, you've raised the level a notch too high.
When you've succeeded at playing the dishwasher or vacuum cleaner concert as loud as the Rolling Stones, now begin with the program we described in our first paragraph -- by using a real dishwasher or vacuum. The tape-recording technique is a good one, but it doesn't work for all dogs since some also fear the vibrations made by such machines.
Desensitization is a tedious effort but it does work.