Heat oil in a large nonstick skillet over medium-high heat. Add chicken; cook 5 minutes on each side or until done. Remove the chicken from pan; cool. Shred the chicken into bite-size pieces. Place chicken, cucumber, and bell pepper in a large bowl; set aside.
Place sugar and the next 6 ingredients (sugar through garlic) in a blender, and process until smooth. Add peanut butter and water; process until smooth, scraping sides. Add peanut butter mixture to chicken mixture; stir well. Add cilantro, and toss well. Warm tortillas according to package directions. Spoon 1/2 cup chicken mixture onto each tortilla; top each serving with 1/2 cup lettuce, and roll up.