We report the first investigation on continuous-wave Raman lasing in high-quality-factor aluminum nitride (AlN) microring resonators. Although wurtzite AlN is known to exhibit six Raman-active phonons, single-mode Raman lasing with low threshold and high slope efficiency is demonstrated. Selective excitation of A$_1^mathrm{TO}$ and E$_2^mathrm{high}$ phonons with Raman shifts of $sim$612 and 660 cm$^{-1}$ is observed by adjusting the polarization of the pump light. A theoretical analysis of Raman scattering efficiency within ${c}$-plane (0001) of AlN is carried out to help account for the observed lasing behavior. Bidirectional lasing is experimentally confirmed as a result of symmetric Raman gain in micro-scale waveguides. Furthermore, second-order Raman lasing with unparalleled output power of $sim$11.3 mW is obtained, which offers the capability to yield higher order Raman lasers for mid-infrared applications.