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 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 an even layer of rice over the nori, leaving a 1-inch border at the top.
Arrange cucumber, carrot, and avocado strips horizontally across the middle of the rice.
Using the sushi mat, carefully roll the sushi, pressing gently to shape it into a tight roll. Seal the edge with a little water.
Repeat with remaining ingredients. Slice each roll into bite-sized pieces with a sharp knife.