I have had this problem since i purchased my jeep about 2 years ago. its a 99 wrangler sport. every now and again when i turn on either my blinkers or whipers my radio will turn off then right back on. its the most annoying thing in the world.
My guess would be loose wiring, but that's pretty tough to track down. Is it a stock radio, or an aftermarket? If aftermarket, you may want to pull it and see how it's wired in, they may have tapped into the blinker for power.
Does it ever blow the fuse out? It's probable a broken or lose wire. You'll just have to get in there with a multimeter and track down where it's coming from.