While I've never enticed one to munch a fly on the main of the river (especially not one of the many breaching ones), I have caught several by fishing them like a redfish on the flats - mostly on the canal. When you see them creating silt puffs as they eat, throw something ahead of them. It works best with silt-colored flies. I use little crab, shrimp, bugger, worm, or crawfish patterns - mostly flies I bought for a redfish trip a while ago - but there are tons of carp-specific patterns out there. I'd think bonefish flies would also work well. Since they're so shallow and most of the flies have bead-heads or weighted eyes, I just use 0x or 2x tippet on a floating line. I understand that mulberry patterns will be in season pretty soon, as well. I have a buddy who uses a popcorn pattern made from white craft foam for common carp, as well.
For what it is worth, after slicing an almost-new fly line to the core, I started tying a short leader section of 30lb mono between fly line and tippet.