Rinse the sushi rice under cold water until the water runs clear.
Combine the rice and water in a rice cooker and cook according to the manufacturer's instructions.
In a small bowl, mix the rice vinegar, sugar, and salt until dissolved. Fold this mixture into the cooked rice.
Place a sheet of nori on the bamboo mat, shiny side down. Spread a thin layer of rice over the nori, leaving a 1-inch border at the top.
Arrange cucumber, carrot, and avocado slices in a line along the bottom edge of the rice.
Using the bamboo mat, roll the sushi tightly from the bottom, applying gentle pressure. Seal the edge with a little water.
Slice the roll into bite-sized pieces with a sharp knife. Repeat with remaining ingredients.