A periodic strip is a finite-width strip of tiles that repeats in one direction with frieze symmetry. They have many potential applications in art and design, particularly because of the ability to construct a finite portion of a periodic strip and wrap it seamlessly around a cylinder. I show that under very mild conditions, shapes that tile the plane also admit periodic strips of any desired width. This fact is true even for aperiodic tile sets. I explain why periodic strips exist, give simple methods for constructing them, and show examples for a variety of well known tilings.