Rinse the sushi rice under cold water until the water runs clear. Combine the rice and water in a rice cooker or pot, and cook according to the package instructions.
While the rice is cooking, mix the rice vinegar, sugar, and salt in a small bowl until dissolved. Once the rice is done, transfer it to a large bowl and gently fold the vinegar mixture into the rice. Let it cool to room temperature.
Place a sheet of nori shiny side down on a bamboo sushi mat. Spread a thin layer of rice on the nori, leaving about 1 inch at the top. Add your choice of fish, cucumber, and avocado along the bottom edge, then roll it up tightly, sealing the edge with water.
Use a sharp knife to cut the roll into bite-sized pieces. Repeat with the remaining ingredients.