Do you have a scrapbooking store nearby? I have used some of the devices with cutters to score the matte board - sometimes they will cut all of the way through if the board is not to thick. Otherwise I use an exacto (sp?) knife to cut through but you need a very steady hand.
You can get a circle cutter which has a pointy end like a pair of compasses does and an adjustable arm with a cutting blade on the end. These work OK on thinnish board but I've never had any luck trying to cut heavy mat board with them. They do have the advantage of being quite cheap though so if you wanted to try one it shouldn't break the bank.
Thanks for the tips. I think my friendly local framer has a device to cut circular and elliptical mounts, so I might ask her to do some for me. I'm sure I can find a use for the mounts too.
How large are the circles that you need to cut? I have a sizzix cutter that has a circle template that would probably cut mat board. It cuts felt and that is pretty much the same thickness of mat board. Harder tho.