Metastructures exhibit vast potential for novel control of elastic wave propagation through careful engineering of their geometry. Recent research has studied such engineered materials and structures that utilize the coupled magneto-mechanical response of magnetoactive elastomers (MAE) to enable adaptive control of dynamic properties. However, MAEs exhibit viscoelastic behavior that strongly influences their frequency-dependent vibration transmission. Here, we computationally study the influence of viscoelasticity on the vibration transmission of an example metastructure using finite element method (FEM) simulations. Frequency-sweep simulations of the metastructure show strong dips in the transmission spectra that are associated with band gaps. A viscoelastic material model is employed, and the loss factor is parametrically varied to study the influence of different damping intensities. Furthermore, the effect of spatially-varying damping on the transmission spectrum is studied through partitioning of the material models used in the FEM simulations. The results show that increased damping causes a smoothing of structural peaks and widening of the transmission trough, with the maximum attenuation unaffected and even enhanced.