Lay a nori sheet on a bamboo sushi mat or flat surface. Spread 1⁄2 cup of the cooked sushi rice evenly over the nori sheet, leaving about an inch uncovered at the top.
In a line across the center of the rice, arrange strips of cucumber, avocado, carrot, and bell pepper.
Using the bamboo mat to assist, carefully roll up the nori sheet tightly around the filling, squeezing gently as you go.
Dampen the uncovered nori edge with water and seal the roll closed.
Slice the roll into 6-8 pieces with a very sharp knife.
Repeat with remaining ingredients to make 4 total rolls.
Serve with pickled ginger, soy sauce, and wasabi on the side.