Rinse the sushi rice under cold water until the water runs clear.
Combine the 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. Cook over medium heat until the sugar dissolves. Let it cool.
Transfer the cooked rice to a large bowl and gently fold in the vinegar mixture. Let the rice 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 a few slices of salmon, cucumber, and avocado along the bottom edge of the rice.
Roll the sushi tightly using the bamboo mat, applying gentle pressure to shape it.
Slice the roll into 6-8 pieces using a sharp knife.
Serve with soy sauce, pickled ginger, and wasabi.