Cut chicken into 1-inch pieces. Place in a self-sealing plastic bag set in a shallow dish. Pour Italian dressing over chicken; seal bag. Marinate in the refrigerator for 2 to 8 hours, turning bag occasionally.
Drain chicken reserving marinade. On 4 medium skewers, alternately thread chicken, sweet pepper, and half of the artichoke hearts, leaving a 1/4-inch space between pieces. Place 2 mushrooms on the end of each skewer. Brush kabobs with reserved marinade.
For a charcoal grill, grill kabobs on the rack of an uncovered grill directly over medium coals for 12 to 14 minutes or until chicken is no longer pink (170 degrees F), turning once halfway through grilling. Brush with 1/4 cup of the pasta sauce during the last 5 minutes of grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place chicken on grill rack over heat. Cover and grill as above.)
Meanwhile, cook pasta according to package directions. Heat remaining pasta sauce and remaining artichoke hearts until heated through. Spoon over individual servings of linguine. Top with kabobs. If desired, sprinkle with Parmesan cheese.