Pick over the crabmeat to remove all traces of cartilage, or cut the imitation crab legs into ½-inch lengths and flake the meat.
Trim off the top and bottom of each radish, then slice them into paper-thin circles. Stack several of these circles on each other and slice them into thin thread-like julienne strips. The strips will be white, tipped with bright red. Soak the radish strips in cold water for a few minutes to help crisp them.
Trim off the stem (darker) end of the cucumber, and rub the cut ends together with a circular motion. A white pasty substance will form around the edge. The Japanese call this aku, or bitterness. Rinse this aku off and pat the cucumber dry. Trim off the other end, then cut the cucumber in half to make two cylinders, each about 2 inches long. Each cylinder will be decoratively cut to yield two “mountains.” Insert the point of a small, sharp knife lengthwise through the middle of one of the cucumber pieces. Keep that knife in place while using a second knife to slash the cucumber on the diagonal. Keeping the first knife in place, flip the cucumber cylinder over and with the second knife make another diagonal slash in the same direction. Remove both knives and pull the cucumber “mountains” apart. Repeat with the other cylinder to make four mountains in all.
In a small bowl, combine the dressing ingredients. Toss the flaked crabmeat in the dressing and allow it to sit for 10-15 minutes. Then, pressing gently against the crabmeat, pour off and reserve the excess dressing.
Drain the radish shreds and toss them in the excess dressing. Allow the radish to sit in the dressing for 5-10 minutes, then drain off the excess.
In each of four glass salad bowls or large brandy snifters, arrange a single green cucumber mountain to the left and toward the back. Divide the crabmeat into four portions, and mound each portion slightly in front and to the right of the cucumber mountain. Divide the radish shreds into four portions and scatter them on top of each portion of crab. Cover and chill for 10 minutes (or for up to 2 hours) before serving.