Bring a large saucepan of lightly salted water to a boil. Add noodles and cook 6 to 8 minutes, or until just tender; rinse briefly in cold water and drain. Transfer to a large serving bowl.
Meanwhile, in a small bowl, with wire whisk or fork, combine peanut butter, water, soy sauce, vinegar, vegetable oil, gingerroot, garlic, sesame oil, and red pepper until blended. Set aside.
Mix noodles with green onion, cucumber, and carrots. Divide mixture among plates and drizzle with sauce.