There are several types of tenkara line: level line (generally fluorocarbon), tapered line (either furled or extruded) and floating fly line are the three most popular. In general, level line is lighter and to achieve the greatest finesse, the lightest possible line should be used. Level line is sold by Japanese line size, which is based on diameter, but effectively is the breaking stength in pounds divided by 4 (size 3 line is 12# test). I would use a size 2.5 or 3 line for any of the rods offered on Finesse-Fishing.com other than the Daiwa LT rods, for which I would use a size 3.5 line.
Tenkara level line is fluorocarbon rather than nylon because fluorocarbon is denser, making the line less susceptible to the wind resistance generated by the cast itself. Nylon, which has less mass, loses too much of its energy to wind resistance. Although fluorocarbon is marketed to bass and trout fishermen as being invisible, tenkara lines are dyed in hi-vis colors so that you can watch the line for strike detection.
There are a few tapered lines that are worth serious consideration by finesse fishermen. Fujino makes a White Tenkara Line, which is visible enough against a dark background like water or streamside rocks and brush, but is much less visible when viewed with the sky as a background. That makes it much more visible to the fishermen than to the fish. It is made from nylon, but it is tapered and the taper does help the line turn over.
Fujino White Tenkara Line
Fujino Soft Tenkara Line (orange) or Tenkara Midi (green)
Fujino also makes hi-vis tapered nylon lines, the Soft Tenkara, which is bright orange, and the Tenkara Midi, which is bright green. In low light conditions they are much more visible than the White line.
The hooks are sharp.
The coffee's hot.
The fish are slippery when wet.