There is another seperate flasher for the turn signals. It is located under the dash, somewhere not too far from the fuse panel. It has only 2 wires going to it and just hangs inside the dash area, not really attached to anything. Replacing that flasher should solve your problem.
I second the burned out bulb, Red_Lx recommended. Just make sure the new bulb is in tight so it doesn't fall out inside the light fixture. Ask me where I keep my spare turn signal bulb!