The one Anne refers to was at a far corner of the right hand side block of the road opposite the PTT - and it was legendary, Fethiye will be a poorer place without it - although I seem to recall it opening close to the Akarca roundabout - maybe all these Birliks are unrelated and it's just a very popular name for a restaurant. The other Birlik is easiest found if you were walking along the sea (Yakamoz) road, where you would turn left for the Tuesday market turn right instead (so just after the sea ends to your right) and you cross a small car park diagonally, turning left when you reach the far left corner of the car park. You are in a street now, walk up and on the corner facing you is another Birlik. Possibly.
There was also a Birlik on Koca Calis Beach of course - that was good too