As an attractive degree of freedom in electromagnetic (EM) waves, the orbital angular momentum (OAM) enables infinite communication channels for both classical and quantum communications. The exploration of OAM generation inspires various designs involving spiral phase plates, antenna arrays, metasurfaces, and computer-generated holograms. In this work, we theoretically and experimentally demonstrate an approach to producing OAM carrying EM waves by a point defect in three-dimensional (3D) photonic crystals (PCs). Simultaneous excitation of two vibrational-defect states with an elaborately engineered phase retardation generates a rotational state carrying OAM. Through converting guided waves in a line defect to localized waves in a point defect and then to radiated vortex waves in free space, the lowest four OAM-mode emitters, i.e., OAM indices of ±1 and ±2, are successfully realized. This work offers a physical mechanism to generate OAM by PCs, especially when the OAM generation is to be integrated with other designs.