Rinse the sushi rice under cold water until the water runs clear. Drain well.

Cook the rice: Combine rice and water in a rice cooker or pot. Bring to a boil, then reduce heat to low, cover, and simmer for 20 minutes. Let it sit for 10 minutes off the heat.

Prepare the seasoning: In a small saucepan, heat rice vinegar, sugar, and salt until dissolved. Gently fold this mixture into the cooked rice and let it cool to room temperature.

Prepare the fillings: Slice the fish into thin strips. Cut the avocado and cucumber into matchstick-sized pieces.

Assemble the sushi: Place a nori sheet on a bamboo mat. Spread a thin layer of rice over the nori, leaving a 1-inch border at the top. Arrange fish, avocado, and cucumber in a line at the bottom edge.

Roll the sushi: Lift the bamboo mat and gently roll the nori over the fillings, pressing lightly to shape. Seal the edge with a bit of water. Repeat with remaining ingredients.

Cut the rolls: Use a sharp, wet knife to slice each roll into 6-8 pieces. Serve with soy sauce, wasabi, and pickled ginger.