Turbine blades and vanes of stationary gas turbines are made of nickel-based superalloys due to their outstanding properties at high temperatures. In order to increase the efficiency and lifetime of turbine components it is of great interest to understand and improve their fatigue behaviour. Especially the turbine blades of the rear stages are conventionally cast for economic reasons. This manufacturing process typically results in formation of a few large grains in the critical component areas. In addition to the reduced number of grains, it is particularly the local elastic anisotropy caused by the crystalline structure, that can lead to high stiffness and stress differences between the grains. Currently used lifetime and design calculation approaches rely on high safety factors to prevent crack initiation within these highly loaded parts. However, this conservative design may often lead to oversized components. In order to fully exploit the material potential, a detailed analysis of the behaviour of a few random orientated anisotropic grains and their influence on each other in highly loaded areas is essentiel. To model random grain morphology Voronoi Tesselation is used to generate cylindrical specimen with 49, 165 and 500 grains, respectively. In order to obtain a statistically random orientation, rotational matrices distributed by the Haar measure, were assigned to the grains and a globally elastic material law resembling the pronounced elastic anisotropy of Ni-base alloys was implemented. Identical total strain controlled loadings were applied to the specimens using the finite element solver ABAQUS, to investigate the materials response for different grain sizes. The results show inhomogeneous stress and strain distribution along the surface and inside the specimen with stress maxima located near grain boundaries for grains with high stiffness differences. Less frequently, stress maxima occur within the grains. Considering the local strains, it can be seen that they can exceed the global strain by a factor of almost two. Furthermore, the elastic anisotropy and the deformation behaviour of surrounding grains leads to multiaxial stress- and strain states within the grains which must be taken into account for a more local lifetime calculation approach.