Cook the sushi rice according to package instructions. While still warm, gently fold in rice vinegar, sugar, and salt until evenly coated. Let it cool slightly to make it easier to handle.
Use a Y-peeler to slice the cucumber lengthwise into thin, flexible sheets. These will act as the outer wrap instead of seaweed. Make sure slices are even to avoid tearing when rolling.
Place a cucumber slice on the bamboo mat, layer with a thin spread of cream cheese, a few slices of avocado, smoked salmon, and a small spoon of sushi rice. Roll tightly using the mat and seal with a little rice.
Use a sharp knife to slice the roll into bite-sized pieces. Wipe the knife clean between cuts for neat edges. Sprinkle furikake or sesame seeds for garnish if desired.