Place the rice and 3¾ cups (935ml) water in a large saucepan. Place over high heat. Bring to the boil. Reduce heat to low. Cook, covered, for 12 mins or until rice is tender and water is absorbed. Set aside, covered, for 10 mins to steam.
Combine the vinegar, sugar and salt in a small bowl. Transfer the rice to a large glass or ceramic bowl. Use a thin wooden spoon or rubber spatula to break up any lumps while gradually adding the vinegar mixture, gently folding to combine. Continue folding the rice for 15 mins or until rice is cool.
Place a sushi mat on a clean work surface with the slats running horizontally. Place 1 nori sheet, shiny-side down, on the mat. Use wet hands to spread one-sixth of the rice over the nori sheet, leaving a 3cm-wide border along the edge furthest from you.
Place a little cucumber, carrot and avocado along the centre of the rice. Holding the filling in place, roll the mat over to enclose rice and filling. Repeat with remaining nori, rice and fillings.
Use a sharp knife to cut each roll into 8 slices. Serve immediately with soy sauce for dipping.