We propose a new, minimal tensorial model attempting to clearly represent the role of microstructure on the viscosity of noncolloidal suspensions of rigid particles. Qualitatively, this model proves capable of reproducing several of the main rheological trends exhibited by concentrated suspensions: Anisotropic and fore-aft asymmetric microstructure in simple shear and transient relaxation of the microstructure toward its stationary state. The model includes only few constitutive parameters, with clear physical meaning, that can be identified from comparisons with experimental data. Hence, quantitative predictions of the complex transient evolution of apparent viscosity observed after shear reversals are reproduced for a large range of volume fractions. Comparisons with microstructural data show that not only the depletion angle, but the pair distribution function, are well predicted. To our knowledge, it is the first time that a microstructure-based rheological model is successfully compared to such a wide experimental dataset. (C) 2018 The Society of Rheology.