Why is my power flickering across multiple circuits?
Sounds like a classic compromised leg of your main service. This problem could be anywhere from the connections at the utility transformer on the pole, to the connection taps on the side of the house, to the meter pan, to the main breaker. 

IMO this is NOT something for a DIY to troubleshoot since you would need to be testing live unfused wires in places that are inaccessible to a homeowner, such as inside the meter pan.

I would start with calling your utility and explaining to them the issue. If they find no problem call an electrician to check the equipment in your house.