Reaction-diffusion mechanics (RDM) systems describe a wide range of practically important phenomena where deformation substantially affects wave and vortex dynamics. Here, we develop the first theory to describe the dynamics of rotating spiral waves in RDM systems, combining response function theory with a mechanical Green's function. This theory explains the mechanically-induced drift of spiral waves as a resonance phenomenon, and it can predict the drift trajectories and the final attractors from measurable characteristics of the system. Theoretical predictions are confirmed by numerical simulations. The results can be applied to cardiac tissue, where the drift of spiral waves is an important factor in determining different types of cardiac arrhythmias.