The magnetoviscous effect, change in viscosity with change in magnetic field strength, and the anisotropy of the magnetoviscous effect, change in viscosity with orientation of magnetic field, have been a focus of interest for four decades. A satisfactory understanding of the microscopic origin of anisotropy of the magnetoviscous effect in magnetic fluids is still a matter of debate and a field of intense research. Here, we present an extensive simulation study to understand the relation between the anisotropy of the magnetoviscous effect and the underlying change in microstructures of ferrofluids. Our results indicate that field-induced chainlike structures respond very differently depending on their orientation relative to the direction of an externally applied shear flow, which leads to a pronounced anisotropy of viscosity. In this work, we focus on three exemplary values of dipolar interaction strengths which correspond to weak, intermediate, and strong interactions between dipolar colloidal particles. We compare our simulation results with an experimental study on cobalt-based ferrofluids as well as with an existing theoretical model called the chain model. A nonmonotonic behavior in the anisotropy of the magnetoviscous effect is observed with increasing dipolar interaction strength and is explained in terms of microstructure formation.