If the front blinkers are working then your turn signal relay is working. You replaced the bulbs so a wire is disconnected somewhere or the bulb housing has corrosion or dirt in it preventing a good connection. First off I'd check the bulb housings for corrosion or dirt then I'd look at the back of the bulb housings to see if a connector isn't hooked up or a wire broke off. If that all looks good you'll need to get a voltmeter to check the circuit.

Ah he beat me too it

Also...
If you have access to another person with a 2G eclipse, see if you can borrow their taillight assembly and plug it up to yours and see if it still doesn't work, which means there is an issue in your tailight wiring harness, if it still doesn't work its an issue on the chassis side.