Does organic chicken need to be cooked differently than conventional chicken?

No, organic chicken cooks the same way nonorganic chicken does.
My favorite way to cook organic chicken is also the simplest—roast chicken. Wash and pat dry a 2–4 pound chicken.
Rub two tablespoons of organic butter into your hands until it is soft and then rub butter all over the inside and outside of the chicken. Salt and pepper the inside of the chicken to your taste and stuff the chicken with half a lemon or orange, half an onion (any type will do), five to ten cloves of peeled garlic, and a small bunch of fresh thyme or rosemary. Salt and pepper the outside of the chicken and roast in a preheated 425-degree oven, breast side down. After about 25 minutes, turn the chicken breast side up for about 40 minutes more, until the chicken is cooked through and juices run clear when you cut between the leg and the thigh. Enjoy!