Cook and leave the rice to stand/steam according to the packet instructions, then transfer to a bowl.
Put the vinegar and sugar in a cup and microwave for 30 seconds until hot. Pour into the cooked rice and mix well – the rice should be sticky. Stir in the sesame seeds.
Lay out the sheets of nori seaweed and divide the rice between them. Spread evenly to the edges, then arrange the avocado wedges and cucumber strips across the length of the rice in the bottom third of each sheet. Starting at the filled end, gently roll the each nori sheet up tightly until a neat roll is formed, brushing with a little water to hold the seaweed tightly in place.
Use a very sharp knife to slice the rolls into 2.5 cm (1 inch) thick pieces. Serve with soy sauce and wasabi for dipping, along with some pickled ginger.