In this work, an analysis of both the mechanical response at the grain scale and high cycle multiaxial fatigue criteria is undertaken using finite element (FE) simulations of polycrystalline aggregates. The metallic material chosen for investigation, a pure copper, has a Face Centred Cubic (FCC) crystalline structure. Two-dimensional polycrystalline aggregates, which are composed of 300 randomly orientated equiaxed grains, are loaded at the median fatigue strength defined at 107 cycles. In order to analyse the effect of the loading path on the local mechanical response, combined tension–torsion and biaxial tension loading cases, in-phase and out-of-phase, with different biaxiality ratios, are applied to each polycrystalline aggregate. Three different material constitutive models assigned to the grains are investigated: isotropic elasticity, cubic elasticity and crystal plasticity in addition to the cubic elasticity. First, some aspects of the mechanical response of the grains are highlighted, namely the scatter and the multiaxiality of the mesoscopic responses with respect to an uniaxial macroscopic response. Then, the distributions of relevant mechanical quantities classically used in fatigue criteria are analysed for some loading cases and the role of each source of anisotropy on the mechanical response is evaluated and compared to the isotropic elastic case. In particular, the significant influence of the elastic anisotropy on the mesoscopic mechanical response is highlighted. Finally, an analysis of three different fatigue criteria is conducted, using mechanical quantities computed at the grain scale. More precisely, the predictions provided by these criteria, for each constitutive model studied, are compared with the experimental trends observed in metallic materials for such loading conditions.