In a microwavable bowl, mix together rice vinegar, sugar, and salt. Microwave in 20-second increments, stirring until sugar has dissolved. Set aside. Rinse rice in a mesh strainer until water runs clear. Place rice and water in a medium saucepan. Bring to a boil over high heat, stir once, then reduce heat to medium-low, cover, and simmer 15 to 20 minutes until water has been absorbed. Immediately transfer rice to a large bowl and cut in vinegar mixture using a large, flat spoon. Allow rice to cool to room temperature, about 30 minutes.
Fill a small bowl with water and lay a sushi rolling mat out flat with a layer of plastic wrap on top. Lay one sheet of nori on top, long side closest to you and rough side facing up. Dip fingers in water and spread about a cup of rice in an even layer across the nori and sprinkle with sesame seeds. Cover with another layer of plastic wrap on top, then flip over so nori is facing up. Set aside top layer of plastic wrap.
In a horizontal row 1 inch from the bottom edge, place a thin row of cucumber, then above it a row of crab, then a row of avocado. Holding onto the closest edge, tightly roll away from you, peeling the mat and plastic wrap away along the way, rolling tightly and pressing down when you reach the other end. Transfer roll to a cutting board.
With a thin, sharp knife, make a quick slice in half, then line the two halves up and quickly slice in thirds, rinsing the knife after each slice. Repeat steps above to make 4 more rolls. Serve with your choice of toppings within an hour of making.
Tips: A mandolin works great for creating thin, even slices of cucumber. Uncoated wooden or bamboo bowls and utensils are ideal for handling the rice, as they absorb the steam while the rice cools.
Per serving: 445 calories, 10 grams fat (2 grams saturated fat), 18 milligrams cholesterol, 691 milligrams sodium, 70 grams total carbohydrates, 5 grams fiber, 12 grams protein.