We demonstrate a framework to understand and predict the far-field emission in terahertz frequency photonic-crystal quantum cascade lasers. The devices, which employ a high-performance three-well active region, are lithographically tunable and emit in the 104-120 microm wavelength range. A peak output power of 7 mW in pulsed mode is obtained at 10 K, and the typical device maximum operating temperature is 136 K. We identify the photonic-crystal band-edge states involved in the lasing process as originating from the hexapole and monopole modes at the G point of the photonic band structure, as designed. The theoretical far-field patterns, obtained via finite-difference time-domain simulations, are in excellent agreement with experiment. Polarization measurements further support the theory, and the role of the bonding wires in the emission process is elucidated.