A functionally graded Boltzmann hierarchical viscoelastic model with both stress- and strain-gradient nonlocalities is developed and implemented to extract results that are more precise than results of Eringen's nonlocal elasticity model. The available size-dependent vibration analyses of the nano/microplates have focused on the frequency analysis and even not the time-dependent transient vibration analyses. In the present research, the forced and transient responses of the microplates are studied comprehensively, for the first time, using a three-element standard solid viscoelastic model. The studied transversely symmetric graded viscoelastic microplate and the relevant function of the material properties variations contain notable hints as well. Furthermore, the resulting new sixth-order nonlocal strain gradient integrodifferential equations are solved by a special technique that includes an analytical spatial Navier series-type solution and a trapezoidal and Runge–Kutta integration schemes, in time domain. Finally, the influences of the stress- and strain-gradient nonlocality parameters and the viscoelasticity parameters on the dynamic behaviors of the viscoelastic FGM microplates are investigated in details. Results show that the effects of the strain gradient nonlocality on the viscodynamic results may be much remarkable than those of the length scale nonlocality, in microscales.