The invention provides shoes and/or shoe elements facilitate natural foot motion and/or reduce forces tending to fight natural foot motion. In at least some such structures, a wearer's heel is secured to the hindfoot region of a shoe (e.g., by a strap system) in a manner that permits heel/forefoot rotation and that allows the lower leg to remain straight. In other structures, a shoe can include a heel supporting component that is separate from a midsole component, and this heel supporting component can move toward the lateral side and/or medial side of the shoe along an interface between the heel supporting component and the midsole component. Other suitable shoe and shoe component structures also are described.