We calculate both the exponent and the prefactor in the nucleation rate of a periodically driven system. Nucleation dynamics is described by the Fokker-Planck equation for the probability distribution of the nuclei over their size. This distribution is found using the concept of the most probable (optimal) nucleation path. The results apply in a broad range of driving force amplitudes, from weak to moderately strong forces where the nucleation rate is changed exponentially strongly, and also in the broad range of the driving frequencies, from low-frequency driving, where the system follows the force adiabatically, to high-frequency nonadiabatic driving. For strong driving forces, the time dependence of the nucleation rate changes from strongly nonsinusoidal to a weak with the increasing frequency of driving. The response of the nucleation rate to the driving force is described in terms of logarithmic susceptibility (LS), which can be obtained from the optimal nucleation path in the absence of the driving. LS is a smooth function of frequency, and therefore even a driving force with comparatively high frequency can change the modulation rate exponentially strongly. LS and the Faraday current are calculated for simple models of electrochemical systems, where the ac driving is produced by modulation of the electrode potential. We also suggest how to find LS from measurements of the average nucleation rate.