Rinse the sushi rice under cold water until the water runs clear. Combine the rinsed rice and water in a medium saucepan. Bring to a boil over medium-high heat, then reduce to low, cover, and simmer for 18-20 minutes or until the rice is tender and water is absorbed. Remove from heat and let it sit, covered, for 10 minutes.
In a small bowl, mix the rice vinegar, sugar, and salt until dissolved. Gently fold this mixture into the cooked rice, being careful not to mash the grains. Allow the rice to cool to room temperature.
While the rice cools, prepare the toppings. Slice the avocado, cucumber, and green onions. Set aside.
Once the rice has cooled, divide it into bowls. Top each bowl with diced tuna, avocado slices, cucumber, shredded carrots, and green onions.
Drizzle soy sauce and sesame oil over the bowls, and sprinkle with sesame seeds. Serve with wasabi and pickled ginger on the side, if desired.