Make the dressing: In a small bowl, whisk together rice vinegar, soy sauce, sesame oil, sugar, and mayonnaise (if using) until smooth. Set aside.
Prepare the cucumber: Slice the cucumber in half lengthwise, then cut into thin half-moons about 1/4-inch thick. Place in a large mixing bowl.
Dice the avocado: Cut the avocado in half, remove the pit, and scoop out the flesh. Dice into 1/2-inch cubes and add to the bowl with the cucumber.
Add the crab: Gently fold in the lump crab meat and sliced green onions, being careful not to break up the crab too much.
Toss with dressing: Pour the dressing over the salad and gently toss until everything is evenly coated. Taste and adjust seasoning with extra soy sauce or vinegar if needed.
Chill (optional): For best flavor, let the salad rest in the refrigerator for 10 minutes before serving.
Serve: Transfer to a serving platter or bowl. Sprinkle with toasted sesame seeds and furikake if desired. Serve cold.