Hyundai: 98 tiburon

Could be:
Turn Signal Fuse
Flasher Fuse (I think this is a possible cause)
Flasher relay
Hazard switch
Other less likely items.

I believe the flasher relay is located in the relay center under the left
hand side of the dash. You'll need to stick your head under the dash to
see the relay center. The flahser is usually the largest relay and
typically has a clip on either side holding it in.

And, to add:
Cycle the hazard switch a few times and see if the turn signals start
working. If they work after this is done, you need a new hazard switch.