Periodic organization of cells is required for the function of many organs and tissues. The development of such periodic patterns is typically associated with mechanisms based on intercellular signaling such as lateral inhibition and Turing patterning. Here we show that the transition from disordered to ordered checkerboard-like pattern of hair cells and supporting cells in the mammalian hearing organ, the organ of Corti, is likely based on mechanical forces rather than signaling events. Using time-lapse imaging of mouse cochlear explants, we show that hair cells rearrange gradually into a checkerboard-like pattern through a tissue-wide shear motion that coordinates intercalation and delamination events. Using mechanical models of the tissue, we show that global shear and local repulsion forces on hair cells are sufficient to drive the transition from disordered to ordered cellular pattern. Our findings suggest that mechanical forces drive ordered hair cell patterning in a process strikingly analogous to the process of shear-induced crystallization in polymer and granular physics.