Stimulated Raman scattering is a third-order nonlinear optical effect that is not only effective for wavelength converting laser output, but also for single longitudinal-mode output due to the absence of spatial hole burning. Diamond is a prominent Raman-active medium that has significant potential for linewidth narrowing and wavelength converting lasers at high power levels due to its high thermal conductivity, long Raman frequency shift and wide spectral transmission range. In this work we utilize diamond in a resonantly mode-matched external cavity to achieve cascaded Raman conversion of a 1064 nm laser. By fine-tuning the length of this external cavity, we can obtain narrow linewidth emission at 1240 and 1485 nm. When operating at maximum power, the measured linewidths were more than twofold narrower than the linewidth of the fundamental field. In addition, the noise levels of the Stokes fields are lower than that of the fundamental field throughout the entire noise frequency range, and the intrinsic linewidth of the second Stokes field, which is expressed at the hertz level (~3.6 Hz), is decreased by approximately three orders of magnitude compared to that of the pump. This work represents the first measurement and analysis of the linewidth and noise characteristics of cascaded diamond Raman lasers and, significantly, offers a new means by which high-power, narrow linewidth laser output can be produced from wavelength-converted laser systems.