High average power lasers with high beam quality are critical for emerging applications in industry and research for defense, materials processing, and space applications. However, overcoming thermal effects in the gain medium remains the key challenge for increasing laser brightness at high powers. Here we report a means for increasing the beam brightness of high-power continuous-wave (CW) beams based on external cavity Raman lasers using diamond, a material with thermal properties far superior to any other laser material. With pump beam quality in the range M2=2.3-7.3, efficient pump-limited conversion to an M2=1.1 Stokes beam is achieved in all cases, with increases in brightness from the pump by factors as high as 12.7. The influence of pump beam quality on laser threshold and slope efficiency is analyzed. This Letter foreshadows an alternative approach for scaling the brightness of CW lasers using high-power, moderate beam quality pumps up to M2=20 or more, such as thin-disk and slab lasers and fiber lasers operating in a mode instability regime.