Electron relaxation, induced by acoustic phonons, is studied for coupled quantum rings in the presence of external fields, both electric and magnetic. We address the problem of a single electron in vertically coupled GaAs quantum rings. Electron-phonon interaction is accounted for both deformation potential and piezoelectric field coupling mechanisms. Depending on the external fields, the ring radii and the separation between the rings, we show that the two different couplings have different weights and importance. Significant oscillations are found in the scattering rates from electron excited states to the ground state, as a function of either the geometry of the system or the external fields.