Rinse the sushi rice under cold water until the water runs clear.
Cook the rice in a rice cooker with the 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. Allow it to cool to room temperature.
Place a sheet of nori on the sushi mat. Spread a thin layer of rice over the nori, leaving a 1-inch border at the top.
Arrange the 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 6-8 pieces with a sharp knife. Repeat with remaining ingredients.