The phase-field method has emerged as a powerful tool for simulating complex cracks in various materials, including composites, because of its ability to naturally model propagation, coalescence, and branching. However, most existing studies are restricted to two-dimensional settings because accurately resolving the small phase-field regularization length scales requires highly refined meshes, making three-dimensional (3D) simulations computationally demanding. In this work, we develop a three-dimensional phase-field framework for modeling thermomechanical fracture in orthotropic composite materials. The framework generalizes the two-dimensional hybrid phase-field formulation of Jain et al. to 3D and extends it to account for coupled thermal effects. A three-field formulation is adopted, in which the equations governing linear momentum balance, transient heat conduction, and phase-field damage evolution are solved to capture thermomechanically driven crack growth. To represent damage anisotropy, a structural tensor is incorporated into the damage evolution equation, enabling direction-dependent fracture behavior consistent with material orthotropy. The proposed method is implemented using the deal.II finite element library, taking advantage of its parallelization and adaptive mesh-refinement capabilities to enable large-scale simulations. The results provide valuable insights about the combined effect of material orthotropy and thermomechanical coupling on complex fracture patterns that arise only in three-dimensional composite fracture behavior.