Place the sushi rice in a large bowl and rinse it several times with cold water until the water becomes mostly clear. Drain thoroughly. This removes excess surface starch and helps produce rice with a pleasant texture.
Add the rinsed rice and measured water to a rice cooker or saucepan. Cook according to the manufacturer's instructions. Once cooked, let the rice rest, covered, for approximately 10 minutes.
Combine the rice vinegar, sugar, and salt in a small bowl, stirring until dissolved. Transfer the warm rice to a wide bowl and gently fold in the vinegar mixture using a rice paddle or spatula. Avoid crushing the grains. Allow the rice to cool until it is close to room temperature.
Cut the salmon or your chosen cooked seafood into long, narrow strips. Slice the avocado and cucumber into similarly sized pieces. Keep the fillings ready beside your sushi mat so you can assemble the rolls efficiently.
Place a bamboo sushi mat on a clean work surface. Cover it with plastic wrap if desired to make cleanup easier. Position one sheet of nori on the mat with its shiny side facing down and its shorter edge closest to you.
With slightly damp hands, spread approximately one-quarter of the seasoned rice over the nori. Leave a strip of about 1 inch uncovered along the far edge. Spread the rice evenly without pressing too hard.
Arrange a small amount of salmon, avocado, and cucumber in a horizontal line across the rice, approximately one-third of the way from the edge nearest you. Avoid overfilling, as too many ingredients can make the roll difficult to close.
Lift the edge of the bamboo mat closest to you and roll the nori over the filling. Use your fingers to keep the ingredients in place while gently tightening the roll with the mat. Continue rolling forward until the uncovered edge seals the roll.
Gently press the roll through the bamboo mat to create an even shape. Use a sharp knife moistened with water to slice the roll into 6–8 pieces. Wipe the blade between cuts for clean edges.
Arrange the sushi pieces on a serving platter. Sprinkle with sesame seeds if desired, and serve immediately with soy sauce, pickled ginger, and wasabi. Enjoy your homemade sushi while the rice and fillings are fresh.