Methods for reconfiguring an atrioventricular heart valve may use systems comprising a partial or complete fenestrated annuloplasty ring proportioned to reconfigure a heart valve that has become in some way incompetent, and a plurality of staples which may have pairs of legs that are sized and shaped for association with the ring at spaced locations along its length. These systems permit relative axial movement between the staples and the ring, whereby a patients heart valve can be reconfigured in a manner that does not deter subtle shifting of the native valve components. Shape-memory alloy material staples may have legs with free ends that interlock following implantation. One alternative is to use flexible rings that will bend in the plane of the ring as the heart beats. Other alternative systems use linkers of shape-memory material having hooked ends to interengage with staples or other implanted supports which, following implantation, decrease in effective length and pull the staples or other supports toward one another so as to create desired curvature of the reconfigured valve. These linkers may be separate from the supports or may be integral with them and may have a variety of shapes and forms. Various of these systems may be implanted non-invasively using a delivery catheter.