How do I quiet noisy drain pipes?
Little bit of sound-proofing knowledge:

People tend to think noise travels in a  straight line from it's source to the reciever.  It does, but a lot of it travels through indirect pathways.

Drive down the road with your window all the way open and listen to the noise levels.  Now drive down the road with your window only open 1/2" and listen.  Now drive down the road with your window fully closed.  Notice how big of a difference there is from just a 1/2" gap!  People put sound deadening on the back of doors all the time and it doesn't work because there is still a 1/2" gap at the bottom.  Use acoustical caulk to seal any openings around the pipes entering your kitchen.

Another source of noise travel is through the rigid studs, floor joists, and ceiling joists because sound travels excellently through these lightweight solids.  They are basically conduits for noise from room to room.  Solid surfaces and objects transfer sound to each other.  You need to isolate your materials.  Go to the dollar store and grab some of those thicker rubber mesh hot pads.  Wrap them around the pipes where they come in contact with anything.  

Fiberglass or mineral wool insulation to the sides of the pipes will do almost nothing.  If you can, get some ROXUL type insulation specific for noise reduction. Squeeze some in between the pipes and the drywall, but do not compress it.  Noise reduction insulation should not be squished so more is not necessarily always better.

... but as @Ecnerwal said cast iron pipes is probably the best option in this circumstance.