The viscosity of suspensions depends on the properties of the fluid, i.e., the matrix material as well as on the volume fraction and shape of embedded inclusions. Regarding this dependence, a micromechanics-based homogenization approach is presented, giving access to the resulting effective viscous properties, starting from well established homogenization schemes originally proposed for the linear-elastic material response. In order to incorporate nonlinear matrix behavior, i.e., non-Newtonian fluids, the secant method of nonlinear homogenization is employed. The effect of nonsphericity of inclusions is considered by substituting the Eshelby tensor by a replacement tensor (replacement Eshelby tensor) determined by linear-elastic finite element simulations in Traxl and Lackner [Mech. Mater. 126, 126-139 (2018)]. In total, suspensions consisting of four different types of matrix materials, namely, (i) Newtonian, (ii) shear thickening, (iii) shear thinning power-law fluids, and (iv) Bingham or Herschel-Bulkley yield-stress fluids are considered embedding spherical, cubical, and tetrahedral inclusions (rigid particles and pores). The performance of the model is assessed by finite element simulations of respective representative volume elements, showing good agreement between model predictions and numerical results for small and moderate inclusion volume fractions.