You have to use a different turn signal flasher. The one you have relies upon the resistance of the filament to cause the bi-metal strip to heat up and open and make the lights blink off. The metal cools and closes to turn the light on again. The constant opening and closing causes the lights to flash for a turn signal and is the source of the noise you hear clicking while the turn signal is on.
With an LED light you need either a computer to count on it's finger and toes to set a fixed period of time to turn on or off (decay counter), or a little motor that rotates with half of a disc set to conduct the other half insulated so the lights are off. this is because LED lights don't offer much in terms of resistance. Almost all of the power is converted into light (you get a max of 18 lumens per Watt out of an incandescent bulb, 70 lumens per Watt out of a fluorescent bulb, and 105 lumens per Watt per watt out of an LED).
Basically you get light instead of heat out of an LED bulb. Who knows may be a an incandescent bulb could save your life if you were buried in an avalanche with those tinny little dash lights acting like heaters.