Remove giblets from the chicken, and season generously with salt. Place in a large pot. Add bay leaves, peppercorns, onion, celery, carrots and turnips (with some of the greens left intact), and potatoes. Fill pot with water until ingredients are 1 inch below the surface, and bring water to a boil.
Reduce heat to low and simmer, skimming foam from the surface, until chicken is cooked through and the vegetables are tender, about 45 minutes. Season to taste with salt and pepper. Stir in dill.
Transfer chicken to a cutting board and let rest 5 minutes. Transfer vegetables to a large serving dish. Carve chicken and arrange with vegetables. Serve immediately, ladling broth over each serving.