I don't think you have to worry about grizzly bears in Carolina (I'm assuming that's the Asheville you're talking about...). The most common bear is the black bear, though it can be of any color, and camping in areas where they live is no big deal. Over here in California, almost every area has the potential for visiting bears, but it's rare.
Keep your food in the provided bear lockers
. Also keep anything else that might have a scent in there. Toothpaste, soap, chewing gum, kleenex. Whatever. Never keep any of that stuff in your car:
If the area you're going to doesn't have lockers or decide to go into the backcountry, get a bear canister
. These can be purchased (or more conveniently, rented) from sporting goods/outdoors stores. You put all your food and anything with a scent in there, lock it up, and keep it at least 50 yards away from your camp. If a bear finds it, it'll bat it around for a while until it realizes it won't get in.
Alternately, you could
hang your stuff in a bag from a tree branch, but it has to be a VERY high branch, so you need a lot of rope to sling the bag over. It's not the most effective way because throwing a bag of food 50 feet in the air isn't always easy, and many bears are smart enough to figure this one out and simply cut the rope. They're pretty damn smart!
Overall, you have nothing to worry about. As long as you don't store any of your food/scented items on you or in your tents, you'll be fine. I've had my food raided by bears twice and while it was a bit nerve wracking, they never came near us. I did have a friend who got his tent rolled by a bear in the Sierras once... he forgot he had beef jerky in his jacket pocket. Oops!