The stress distribution in a random polycrystalline material (Alloy 600) was studied using a topologically correct microstructural model. The distributions of von Mises and hydrostatic stresses, which could be important factors when studying the intergranular stress corrosion cracking, at the grain vertices were analysed as a function of microstructure, grain orientations and loading conditions. The grain size, shape, and orientation had a more pronounced effect on stress distribution than the loading conditions. The stress concentration factor was higher for hydrostatic stress (1.7) than for von Mises stress (1.5). Hydrostatic stress showed more pronounced dependence on the disorientation angle than von Mises stress. The observed stress concentration is high enough to cause localized plastic microdeformation, even when the polycrystalline aggregate is in the macroscopic elastic regime. The modelling of stresses and strains in polycrystalline materials can identify the microstructures (grain-size distributions, texture) intrinsically susceptible to stress/strain concentrations and justify the correctness of applied stress state during the stress corrosion cracking tests.