Platinum-coated Janus colloids exhibit self-propelled motion in aqueous solution via the catalytic decomposition of hydrogen peroxide. Here, we report their motion in a uniformly aligned nematic phase of lyotropic chromonic liquid crystal, disodium cromoglycate (DSCG). When active Janus colloids are placed in DSCG, we find that the anisotropy of the liquid crystal imposes a strong sense of direction to their motion; the Janus colloids tend to move parallel to the nematic director. Motion analysis over a range of timescales reveals a crossover from ballistic to anomalous diffusive behavior on timescales below the relaxation time for liquid crystal elastic distortions. Surprisingly we observe that smaller particles roll during ballistic motion, whereas larger particles do not. This result highlights the complexity of phoretically-driven particle motion, especially in an anisotropic fluid environment.