My experience (gained before I gained status levels at AC) is that if you ask nicely, are generally polite, and the plane is not full, the check-in agents will usually accept one additional piece of checked baggage without charging you for it.

Assuming they are normal sized pieces, of course, and not 3 of those Asian special suitcases that you can pack a 20 foot container in.

What you do have to be careful about is the weight of the pieces. You must never exceed 70 pounds per piece. If you do, the item must be shipped as cargo, because it presents an occupational hazard to the ground staff to attempt to handle baggage in excess of 70 pounds by hand.