Up to 1/4" thick and 6" diameter plywood I use a circle cutter that can be found at WoodCraft or Rockler that fits on my drill press. I do have the thickness on one side then flip the board for the other side. With a cicle cutter I can do both the inside and outside cuts of a centering ring. Beyond those dimensions I use a circle cutter jig that can be purchased from Highland Woodcraft.
I have used a router before and was pleased with the finish. What I don't like about using a router is that you must either do a rough cut or accept quite a bit of waste due to the router bit thickness. Since I've got the circle cutting jig, I just do the entire outer cut on the band saw.
Of late, I've been using large circle cutting bits for the interior cut of centering rings since I don't have good enough technique on my scroll saw to cut a good circle. The interior cut is a good place for a router if you have a pattern and have already done a rough cut.
I'm open to instruction. What do the rest of you do?