We discuss a new approach for measuring flaring activity cycles of blazars and quasars. The activity cycles of 21 active galactic nuclei were estimated using the radio total flux-densities at five frequencies (5 GHz-37 GHz) and high resolution long-term very long-baseline interferometry (VLBI) observations (1.6 GHz-43 GHz). We define the activity cycles as the time period between two successive "core" outbursts in the total flux-density radio light curves. The "core" outbursts show frequency-dependent time-delays, flat amplitude spectra and are associated with brightening of the VLBI core. We show that the activity cycles are very long and have durations from 4 to >25 years in observer's frame. We also show that a few sources, such as S5 1803+784, reveal less active behavior if we take into account the long-term VLBI and total flux-density observations for more than 20 years. We found for four sources that core flares appear periodically and that activity cycles coincide with a period in the total flux-density variability. Analysis of the activity cycles leads us to suggest that more massive blazars and quasars possess shorter activity cycles. Quasars and blazars with mass as of about 10⁸ M_sun have activity cycles of ∼12 yrs, whereas sources with mass as of 10¹⁰ M_sun have activity cycles of about 2 yrs in the source frame.