Combine rinsed sushi rice and water in a saucepan. Bring to a boil over medium-high heat, then reduce heat to low, cover, and simmer for 10 minutes until the water is absorbed and rice is tender. Remove from heat and let it steam for 10 more minutes.
In a small bowl, stir together rice vinegar, sugar, and salt until the sugar and salt are dissolved. Pour this mixture over the cooked rice in the mixing bowl, gently folding to evenly coat the grains. Allow the rice to cool slightly while you prepare other ingredients.
Arrange the sliced cucumber and avocado on the serving bowls. Slice the avocado into thin, even pieces and place over the rice or around the bowl for visual appeal.
Scoop the cooled rice into bowls, creating a base layer. Top with the sliced cucumber and avocado, then sprinkle evenly with toasted sesame seeds for crunch and flavor.
Serve immediately with additional seasonings or soy sauce if desired. The dish should have a firm, slightly sticky rice base with crisp vegetables and a nutty sesame topping.