A five or six weight with a floating line works for shallow edges; add an intermediate or sinking line when the cover lies deeper. Choose a leader that will turn over the fly and tolerate the structure you are fishing.
A small Clouser, Bendback or shrimp gives you several ways to explore the same snag. Let the fly descend beside the cover, then try a short pull and a distinct pause. Many fly-fishing accounts describe takes on the drop. Keep the rod pointed along the line and set with a short strip when the weight comes on.
After catching a fish, cover the area again. Perch can school, although disturbance may put them off. If the bank goes quiet, give it a rest and return from a different angle rather than repeatedly landing the line over the fish.