Rinse the sushi rice under cold water until the water runs clear.
Combine rice and water in a rice cooker and cook according to the manufacturer's instructions.
In a small saucepan, combine rice vinegar, sugar, and salt. Heat until sugar dissolves. Let cool.
Transfer cooked rice to a large bowl. Gradually fold in the vinegar mixture until rice is well-coated. Let cool to room temperature.
Place a sheet of nori on a bamboo sushi mat. Spread a thin layer of rice over the nori, leaving a 1-inch border at the top.
Arrange cucumber, avocado, and fish in a line along the bottom edge of the rice.
Roll the sushi tightly using the mat, applying gentle pressure. Seal the edge with a little water.
Slice the roll into 6-8 pieces with a sharp knife. Repeat with remaining ingredients.
Serve with soy sauce, pickled ginger, and wasabi.