We present a scheme to solve three-dimensional viscoelastic anisotropic wave propagation on structured staggered grids. The scheme uses a fully-staggered grid (FSG) or Lebedev grid (Lebedev, J Sov Comput Math Math Phys 4:449–465, 1964; Rubio et al. Comput Geosci 70:181–189, 2014), which allows for arbitrary anisotropy as well as grid deformation. This is useful when attempting to incorporate a bathymetry or topography in the model. The correct representation of surface waves is achieved by means of using high-order mimetic operators (Castillo and Grone, SIAM J Matrix Anal Appl 25:128–142, 2003; Castillo and Miranda, Mimetic discretization methods. CRC Press, Boca Raton, 2013), which allow for an accurate, compact and spatially high-order solution at the physical boundary condition. Furthermore, viscoelastic attenuation is represented with a generalized Maxwell body approximation, which requires of auxiliary variables to model the convolutional behavior of the stresses in lossy media. We present the scheme’s accuracy with a series of tests against analytical and numerical solutions. Similarly we show the scheme’s performance in high-performance computing platforms. Due to its accuracy and simple pre- and post-processing, the scheme is attractive for carrying out thousands of simulations in quick succession, as is necessary in many geophysical forward and inverse problems both for the industry and academia.