Rinse the rice in a fine-mesh strainer until the water runs mostly clear, then drain well.
Combine rice and water in a saucepan or rice cooker. Cook according to the rice cooker instructions, or bring to a boil, cover, and simmer on low for 12 to 15 minutes. Let stand covered for 10 minutes.
Warm the rice vinegar, sugar, and salt in a small pan just until dissolved. Fold the seasoning into the hot rice with a slicing motion, then spread the rice on a tray to cool until barely warm.
Place one nori sheet shiny-side down on a bamboo mat. With damp fingers, spread one-eighth of the rice over it, leaving a 1-inch strip uncovered at the far edge.
Arrange shrimp, cucumber, and avocado in a narrow line across the center. Lift the near edge of the mat and roll tightly away from you, using gentle pressure to shape the roll. Moisten the exposed nori strip to seal.
Repeat with the remaining 7 sheets and filling to make 8 rolls. Cut each roll into 6 pieces with a damp, sharp knife, wiping the blade between cuts. Serve with soy sauce, pickled ginger, and wasabi.
