The response of turbulence to a periodic forcing of the Reynolds number is studied. The turbulent flow is produced in a closed geometry between two counter-rotating disks. The time series of both the global injected power and the local velocity are measured. It is found that the injected power fluctuations exhibit resonances for well-determined values of the modulation frequency, making it possible to estimate a turbulent cascade time. For modulation periods larger than the measured cascade time, the response of the velocity fluctuations is simply proportional to the modulation of the forcing. For high-frequency modulations, the velocity fluctuations are strongly damped: their amplitude decreases as the inverse of the modulation frequency.