The magnetic field dependence of the energy and linewidth of the transition from the n=1 to the n=0 Landau level of a piezoelectric polaron has been calculated numerically for polarons at zero temperature. A weak isotropic piezoelectric coupling between the electron and the acoustic phonon modes is assumed, and is treated as a perturbation on free-electron magnetic eigenstates. It is found that the shift in the cyclotron resonance frequency due to piezoelectric electron-phonon interaction begins to differ drastically from that expected from the polaron effective-mass theory when ℏωcmc2>1, where ℏωc is the separation in energy of the unperturbed magnetic levels, m is the band mass of the electron, and c is the velocity of sound in the crystal. The semiclassical theory of Mahan and Hopfield is reviewed and shown not to be suitable for interpreting recently reported cyclotron-resonance experiments in CdS, where the Landau-level spacings were substantially greater than the mean thermal energy per electron. Difficulties encountered in extending the present perturbation calculation to finite temperature are pointed out. Finally, the weak-coupling energy shift of the n=0 to n=1 transition for optical polarons (electrons coupled to longitudinal optical phonons) is evaluated as a function of magnetic field and compared to previous results derived for weak fields. It is suggested that the markedly nonlinear magnetic field dependence of the energy shift found might offer an attractive experimental way of observing optical-polaron effects on the electron self-energy.