Fracture propagation in engineering materials can be anisotropic, with fracture toughness varying with direction due to material microstructure, manufacturing processes, or loading history. These materials exhibit preferred directions of propagation where the fracture toughness, and consequently, the critical energy release rate, is reduced. This behavior is observed in composite materials, layered geological formations, and additive-manufactured components. Accurately predicting fracture propagation in such materials is therefore of significant importance for reliable analysis and design. This work presents a 3D Generalized Finite Element Method (GFEM) for the simulation of fracture propagation in materials with non-isotropic fracture toughness. With the GFEM, enrichment functions capture discontinuities independently of the underlying finite element mesh, enabling robust simulation of complex fracture paths without the need for remeshing. Fracture growth is governed by the Generalized Maximum Energy Release Rate (GMERR) criterion, in which the crack advances in the direction that maximizes the ratio between the computed energy release rate and the orientation-dependent critical fracture energy. A method based on the golden-section algorithm is proposed to compute the direction that maximizes this quantity. Examples are presented to demonstrate the method’s ability to capture anisotropy-driven fracture propagation and to show that the proposed GFEM framework provides an accurate, robust, and efficient tool for such simulations.