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 saucepan, heat the rice vinegar, sugar, and salt until dissolved. Let it cool.
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 bamboo mat. Spread a thin layer of rice over the nori, leaving a 1-inch border at the top.
Arrange the cucumber, carrot, avocado, and fish in a line along the bottom edge of the rice.
Roll the bamboo mat away from you, pressing gently to shape the roll. Seal the edge with a bit of water.
Slice the roll into 6-8 pieces with a sharp knife. Repeat with remaining ingredients.