Japanese tenkara techniques influenced the way the gear developed, or in other words, the gear we have now was designed specifically to fish the Japanese tenkara techniques. You can certainly use it to fish in other ways, but the more you use a tenkara rod the more you are likely to realize that it was exquisitely designed for a certain purpose.
Tenkara as we know it today was developed to catch trout (amago and yamame) and char (iwana) in small mountain streams. The fish are not very large, reaching no more than about 15", and most are much smaller than that. Because both the streams and the fish are small, there was never a need for long casts or reels to hold excess line. A long bamboo pole and horsehair line was sufficient.
As in the West, the first Japanese flies were wet flies. In Japan, however, there doesn't seem to have been the equivalent of an F. M. Halford, who believed flies should be exact imitations of the natural insect, and fly fishing should be upstream, with a dry fly, and only to a rising fish. Through his extensive writings, Halford had a commanding influence on a generation of fly fishermen, and has a lingering influence even to this day. The Japanese of Halford's day fished for food, not for sport, and the tenkara techniques they developed were simple, efficient and effective. In Japan even now, most tenkara fishing is done with wet flies.
The typical Japanese tenkara technique is to fish a single wet fly perhaps 2 to 5" under the surface. Unlike most wet fly fishing in the US, they are generally not fished "down and across," but upstream, quartering upstream, or directly across. Drifts tend to be relatively short, as one cannot strip in line, and must raise the rod tip to keep the line tight. As the rod gets too high to allow an effective strike if a fish takes the fly, the fly is picked up and another cast is made.
I have noticed that a large percentage of my fish are hooked just as I pick up the fly to make another cast. As you pick up to make a new cast, the action of a wet fly ascending to the surface seems to be a powerful trigger causing fish to strike. Fishing a wet fly with short drifts means you pick up more frequently, providing more triggers to more fish. At all times, the rod is kept relatively high to keep the line relatively tight and off the water. With the end of the line off the water (only the tippet and fly in the water) it is significantly easier to identify strikes. And with a tighter line, it is much easier to hook the fish that do strike.
Japanese wet flies are not weighted, and to allow a deeper drift, it is necessary to cast further upstream from where the fish are expected to be, and the line must be a bit slack to allow the fly to sink. The delicacy of presentation that I frequently mention is really more applicable to American tenkara techniques than traditional Japanese tenkara techniques. When fishing a wet fly, it is often necessary to almost slap the fly (not the line) down so that it breaks through the surface quickly. Delicacy is for dries.
Although much is written (including by me) about achieving a drag-free drift, it is often more productive to manipulate the fly to give it the impression of life. In the US, the most common or at least most written about manipulation is the "Leisenring Lift." Of all the Japanese tenkara techniques, the one most written about is surely the technique that translates into English as the "invitation." The invitation is a rhythmic pulsing of the wet fly that is done by very slightly raising and lowering the rod tip. This is done most commonly with the sakasa kebari or reverse hackled flies (flies in which the hackle slants forward over the hook eye rather than backwards toward the hook bend).
As the rod tip is raised and the line is tightened, the fly is pulled forwards and upwards slightly, causing it's soft hackle to bend backwards. As the rod tip is dipped slightly, the fly sinks a bit further in the water, and the hackle opens up and bends forward. This rhythmic pulsing of the hackle makes the fly look like it is alive, and it draws or "invites" strikes. The pulses should be very subtle, though, as the rod tip only has to move a few inches to take up the slack and open and close the hackle, and it only takes the slightest movement to suggest life. We're talking about a struggling insect here, not an Olympic swimmer. Also, if you tug the line to the point that the line "bounces" around, you'll find it much harder to detect strikes. Although strikes to a pulsing Sakasa Kebari tend to be more aggressive, not all of them are. Your pulses should be enough to open and close the fly's hackle, but not enough to make your line bounce.