Tenkara lends itself very well to sub-surface angling techniques. Tenkara is also more useful on generally open streams, while I have found Western fly fishing techniques (which is more complicated) to be more useful on tiny thick, brushy streams. But it all depends upon the skill of the angler. Try using shorter line lengths on your Tenkara rod for those overgrown waters.
I personally lean toward wool body flies for this type of fishing. Generally you see smaller size #16-18 and darker flies like black or dark brown, and mohogany early in the season, changing to larger size #14-12 and brighter insects, like a medium brown, olive, tan or creamy colors in summer. Fall brings terrestrials like ants, beetles and grasshoppers into focus.
As far as sizes are concerned, it partially depends upon the sizes of the waters that you wish to fish, but generally a range from size #12 and smaller is fine. Size #14 is probably the most common insect size, but I would also want to add a few #16s to your selection as these are often seen too.
Certain streams in the west see large numbers of very tiny insects like a size #18-22, especially mountain waters. But, you probably only need a couple of these unless you find yourself faced with needing to fish these sizes all of the time. In lieu of sampling the stream bed, to find out you just change flies over time until you find what works best - most of the time.
It doesn't have to be complicated at all. A few simple choices as to color and a small range of sizes are sufficient. After you have some time in on the water, you can add to or change your selection as needed. Generally, the most common colors to select are brown, black, olive, gray and tan. It also doesn't hurt to keep a larger bright fly on hand, like a size #10 red to try if nothing else works, or on a fast water stretch.
Anyway, keep your selection small and simple at first. As your knowledge and experience grows, so can your fly selection. Not knowing where you will be fishing or on what streams, we could be more careful of your fly selection if we knew that.
I recommended this selection to a customer recently. Generally, nationwide, and for sub-surface fishing I would start with the following, keeping it minimal:
4- Killer bugs (weighted+) of some type in size #14
2- Dark (weighted) Killer Buggers (black, brown etc.) size #12
2- Japanese style wool body soft hackle (like a gray, brown or olive etc.) in size #14 or 16
1- Large, bright floating fly in size #10 or 12, red or orange is good
1- Dark midge size #18 or 20
1- Worm type fly for rising water
1- Bushy Wooly Bugger in brown or olive (imitating crayfish or sculpins).
That's a dozen flies and would certainly get you started in most places. The killer bugs I would select based upon fishing either East, or West in the US.
Really, more important than fly selection is how you fish. A careful approach to the water, observing for a bit before putting your lure in is important. Keep low and keep your shadow off the water at all times. You will often find that most of the fish that you see are uncatchable, simply because they will see you coming and hide. That's why most anglers prefer to fish upstream since the fish face this way. However, they are very observant so be cautious. The more time you get on the water, the more effective that you will be, more of the time.