► A small-strain FFT-based model for elasto-viscoplastic polycrystals is presented. ► Local fields and effective behavior can be computed from a microstructure image. ► Implicit time discretization and an augmented Lagrangian iterative scheme are used. ► Benchmarks for mechanical behavior, boundary conditions and convergence are presented. ► The model is used to study the influence of crystal anisotropy on stress hot-spots. We present the infinitesimal-strain version of a formulation based on fast Fourier transforms (FFT) for the prediction of micromechanical fields in polycrystals deforming in the elasto-viscoplastic (EVP) regime. This EVP extension of the model originally proposed by Moulinec and Suquet to compute the local and effective mechanical behavior of a heterogeneous material directly from an image of its microstructure is based on an implicit time discretization and an augmented Lagrangian iterative procedure. The proposed model is first benchmarked, assessing the corresponding elastic and viscoplastic limits, the correct treatment of hardening, rate-sensitivity and boundary conditions, and the rate of convergence of the numerical method. In terms of applications, the EVP–FFT model is next used to examine how single crystal elastic and plastic directional properties determine the distribution of local fields at different stages of deformation.