The nonlinear oscillations of viscoelastic microplates is addressed in this paper based on the modified couple stress theory (MCST). Employing the Kirchhoff plate theory, both out-of-plane and in-plane motions as well as the corresponding inertia are taken into account; an internal damping mechanism, based on Kelvin–Voigt model, is employed to model the behaviour of the material. The strain energy, the kinetic energy, the work done by the viscous parts of the classical and non-classical stresses, and the work of the external time-dependent force are obtained and implemented into Hamilton's framework so as to derive a set of fully coupled nonlinear partial differential equations (PDEs) for motions in the out-of-plane and in-plane directions. The Galerkin scheme is utilized to reduce the set of viscoelastically coupled nonlinear PDEs into a set of nonlinear ordinary differential equations (ODEs). Thereupon, this set of equations is transformed into a new set of size-dependent viscoelastically coupled nonlinear first-order ODEs and then is solved with the aid of a continuation scheme. The nonlinear oscillations is thoroughly investigated through conducting extensive numerical simulations and plotting force-response and frequency-response diagrams of the viscoelastic microsystem. The results reveal that the contributions of the nonlinear damping terms, arising due to employing a viscoelastic model, in the response of the viscoelastic microsystem increase substantially when the forcing amplitude is increased. Moreover, the concurrent presence of the nonlinear amplitude-dependent damping mechanism and the length-scale parameter affects the resonant response of the microplate significantly in both linear and nonlinear senses.