The anisotropic mechanical behaviour of additively manufactured (AM) lattice structures is governed not only by their geometric design but also by the crystallographic texture of the base material, which is directly influenced by the manufacturing process. This study investigates the joint effect of geometric and texture-induced anisotropy in AM lattice structures using crystal plasticity finite element method (CPFEM) simulations. To enable such analyses, a Python-based tool is developed within the Abaqus/CAE scripting interface for generating polycrystal FE models with user-defined grain size, morphology and texture. This algorithm facilitates CPFEM simulations of complex geometries, allowing for a comprehensive analysis of the relationship between microstructure and mechanical behaviour. With particular application to lattice structures, it allows for the study of the evolution of Young's modulus anisotropy across various strut diameters and relative densities, as well as the construction of effective yield surfaces. Results show that changes in crystallographic texture can significantly alter both the elastic and plastic response of the lattice materials, affecting stiffness, yield strength, and deformation characteristics. Neglecting crystallographic texture effects can result in considerable errors in predicting mechanical properties, highlighting the importance of texture-aware modelling for AM lattice structures.