Another method is to use a Dremel tool with a small pointed stone to grind down the outer edges of the "rivet" to until the mushroomed head is thin enough to allow the medallion to be pressed free of the washer with a pin punch.
It's a bit more tedious, but I've found that it leaves a tad bit more of the stud end intact to make it easier to re-install and re-flatten the end so it looks like factory and not cave in the washer so much.
I've also used a small bit of epoxy on the underside of the washer to afix it to the wood and medallion stud, it doesn't show. I let it fully set up and then "round off" the end of the stud only enough to make it look like original, letting the epoxy do the job of keeping everything together.
I use a nail set punch that's a larger diameter than the stud, some nail set punches has a recess in the tip for the finishing nail head to ride in to keep the punch from slipping off and it's just right for "re-riveting" the end of the medallion stud to the washer.
