With my new washer, I've noticed that when the machine switches mode like from wash to rinse, the lights in the kitchen and dining room seem to flicker or dim ever so slightly. And I have to wonder what is causing this?

Sounds like the washer is on the same circuit. You might want to look at your breaker box and double check the washer circuit. If it did it to the whole house then I would have the electic company out to check your service amperage, but since you say its only affecting a couple lights then it sounds like its on that same circuit.