Halve avocado. Using the end closest to the handle of the knife, swiftly wedge the blade into the pit; turn knife with pit slightly and pull out. Using the tip of the knife, slice avocado halves vertically and then once horizontally down the middle. Scoop out the sliced halves using a spoon. Alternatively, you can scoop out whole halves and then mash the avocado.
Tear imitation crab into pieces and place in a bowl; season with mayonnaise and salt. Cut cucumber into 1-inch-long julienne.
Cover a bamboo rolling mat with plastic wrap. Put a sheet of dried seaweed on top of the plastic. Spread 1/4 of sushi rice on top of the nori sheet. Sprinkle sesame seeds over the sushi rice.
Using the plastic wrap, turn the sushi layer over so that the seaweed is on top. Remove the plastic wrap and place avocado, crab and cucumber horizontally on the seaweed. Roll the bamboo mat toward you, pressing the ingredients inside the cylinder-shaped sushi. When finished rolling, press firmly on the bamboo mat with hands, unroll the mat and then remove the rolled sushi.
Repeat to make more rolls. Wipe a knife with a wet cloth before slicing sushi. Cut the sushi roll into bite-size pieces.