Rinse the sushi rice under cold water until the water runs clear. Place the rice and water in a pot, bring to a boil, then reduce heat and simmer covered for 15 minutes. Remove from heat and let it sit covered for 10 minutes.

Mix the rice vinegar, sugar, and salt in a small bowl until dissolved. Spread the warm rice in a large bowl and gently fold in the vinegar mixture. Let the rice cool to room temperature.

Lay one nori sheet on a bamboo mat. Spread about 3/4 cup of seasoned rice evenly over the nori, leaving a 1-inch border at the top. Arrange cucumber matchsticks, carrot matchsticks, and avocado slices in a line across the center. Sprinkle a few black sesame seeds over the fillings.

Roll the sushi tightly from the bottom edge, pressing gently as you go. Seal the top edge with a little water. Repeat with the remaining nori sheets and fillings.

Use a sharp knife to slice each roll into 6 pieces. Arrange the slices so the vegetable strips are visible.