Bluegill were made for fly fishing. They are one of my favorite fish. What great fighters! Especially those big males. I'll just make a list of what I know from here in the North. My suggestions may or may not be strictly Tenkara style, or useful for you, but are what work for me and flies that have proven popular here in the store. Are you fishing for them in deeper water? Or when they are inshore defending nests? Water temperature is key. They move inshore when the water reaches about 70 degrees F.
Reason I ask is... if they are on beds, you can catch them on anything and almost nothing. They vigorously defend their nests chasing and attacking anything coming near. And as a naturalist, you should not bother them too much so that the catch-able fish populations won't be affected.
When in deeper water, Bluegill feed on aquatic insects (which species is present is largely dependent upon the water depth, cleanliness and composition of the bottom). They also love freshwater shrimp when available and tiny minnows bring up the last chosen item. At this time a tiny streamer, killer bugger, killer bug or a wool body kebari like the Killer Kebari in general tan/brown color is a good bet. Fish the fly slowly and deliberately to attract the largest fish. Best general color combinations here in the North, in my experience are: Brown and Orange like the Copperbari, Grey and Yellow, like the Takayama and Brown and Black, like the Dr. Ishigaki.
In the late summer & fall Bluegill will cruise the shoreline looking for terrestrial insects and the midges that are common there. An ant, black kebari or #16 Ishigaki Grizzly or Elk hair Caddis are good bets. A good, general prospecting fly is a small, slow sinking type, such as the brass or copper weighted wet fly or nymph pattern as opposed to a jig head.
Speaking of jig heads, we need to mention deep water. Here is a little known secret about fishing for deep water fish from a boat - Predator fish key in on the movements of prey fish. Prey fish rise straight up toward the surface to try to shake off the predator. So fish a white jig head fly straight up and down, or slightly ahead of a moving boat, lifting the fly straight up when the boat comes along side. You can manage the same thing from the
bank, in deep stream pools fishing for trout.
Bluegill move to deep water in summer and winter and can be caught suspended over open, aquatic vegetation or brushy water structures by using weighted flies. #10-14 wet flies and nymphs, Killer Bugs and Killer Buggers. Fish these flies using long, slow, deep retrieves.
Bluegill eat the same foods as trout: insects, tiny crustacean, worms, leeches, and small minnows, and definitely in that order. The general idea in angling is to represent the most fish-popular, abundant natural foods with flies. But, you should err on the side of smaller size flies when it comes to Bluegill. They seem to prefer the smaller offerings. Think size #10-18. Also I would go for barb-less on the hooks if possible for easy release, for both you and the fish.