Rinse the sushi rice under cold water until the water runs clear.
Cook the rice in a rice cooker with 2.5 cups of water.
In a small bowl, mix the rice vinegar, sugar, and salt until dissolved.
Once the rice is cooked, transfer it to a large bowl and gently fold in the vinegar mixture. Let it cool to room temperature.
Place a nori sheet on the sushi 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 strips in a line along the bottom edge of the rice.
Using the sushi mat, roll the nori tightly around the fillings. Seal the edge with a little water.
Slice the roll into bite-sized pieces with a sharp knife. Repeat with remaining ingredients.