In a small bowl, toss the warm cooked sushi rice with rice vinegar until well combined and slightly sticky.
Slice the sushi-grade salmon, cooked shrimp, avocado, and cucumber into long, thin strips for easy rolling.
Place one nori sheet on a bamboo rolling mat and spread half of the rice evenly over the bottom two-thirds of the sheet.
Arrange half of the salmon, shrimp, avocado, and cucumber strips in a horizontal line across the center of the rice.
Carefully lift the edge of the mat and roll the nori tightly over the fillings, using a dab of water to seal the edge.
Repeat with the second nori sheet and remaining ingredients, then use a sharp, wet knife to slice each roll into six even pieces.
Serve the fresh rolls immediately with tamari, wasabi paste, and pickled ginger on the side.