Rinse the sushi rice in a fine-mesh strainer under cold running water until the water runs clear.
In a medium saucepan, combine the rinsed rice and 2 cups of water. Bring to a boil, then reduce the heat to low, cover, and simmer for 15 minutes or until the rice is tender and the water is absorbed.
In a small saucepan, heat the rice vinegar, sugar, and salt over low heat until the sugar and salt dissolve. Remove from heat.
Transfer the cooked rice to a large wooden or glass bowl. While it's still hot, drizzle the vinegar mixture over the rice and gently fold it in with a wooden spatula. Fan the rice as you mix to cool it down and achieve a glossy finish.
Slice the avocado and julienne the cucumber into thin strips. Have them ready for assembly.
Place a bamboo sushi rolling mat on a clean surface and cover it with plastic wrap. Lay a sheet of nori, shiny side down, on the mat.
Wet your fingers to prevent sticking, then spread a thin layer of sushi rice evenly over the nori, leaving about half an inch at the top.
Arrange avocado and cucumber slices in the center of the rice-covered nori.
Carefully lift the edge of the mat closest to you and begin rolling the nori, rice, and filling away from you, applying gentle pressure to shape the roll.
Wet the exposed nori edge with a bit of water to seal the roll.
Continue rolling until you have a tight cylinder. Repeat this process with the remaining nori sheets and ingredients.
Serve the sushi rolls with soy sauce, pickled ginger, and wasabi (if desired).
