We report on the first demonstration of an all-fiber CW Raman laser based on a multimode graded-index fiber directly pumped by multimode fiber-coupled laser diodes. A joint action of Raman clean-up effect and mode-selection properties of special fiber Bragg gratings inscribed in the central part of the graded-index fiber core, results in high-efficiency conversion of a multimode (M2~26) pump at 915 nm into a high-quality (M2~2.6) output beam at 954 nm. About 50 W output power has been obtained with slope efficiency of 67%. The proposed development and integration of key multimode fiber technologies opens the door to new type of LD-pumped high-power high-beam-quality fiber lasers that may operate at almost any wavelength defined by available LDs.