Sounds like your microwave and audio system are sharing the same circuit. Less than ideal, but the temporary voltage drop should not cook your audio gear. If this is prolonged, and your audio gear has a weak power supply, then it could cause premature failures of sensitive integrated circuits (chips).
Sonically, I would expect you would hear it, and that would annoy me to no end. You could try AC regenerators, but the real solution is that both the microwave and your audio system should be on independent dedicated electrical circuits.