Rinse the sushi rice under cold water until the water runs clear.
Cook the rice according to the package instructions.
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 sheet of nori 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 along the bottom edge of the rice.
Using the sushi mat, roll the nori tightly around the fillings. Moisten the top border with water to seal the roll.
Slice the roll into 6-8 pieces with a sharp knife. Repeat with remaining ingredients.