Grantham was born in Duncan, Oklahoma. After receiving a Bachelor of Music from the University of Oklahoma, he went on to receive his MM and DMA from the University of Southern California. For two summers he studied under famed French composer and pedagogue, Nadia Boulanger at the American Conservatory in France. His music has won many prestigious awards, including the Prix Lili Boulanger, the ASCAP Rudolf Nissim Prize, and First Prize in the National Opera Association's Biennial Composition Competition. Grantham is the recipient of a Guggenheim Fellowship and three separate grants from the National Endowment for the Arts. The symphony orchestras of Atlanta, Cleveland, and Dallas are among the ensembles that have commissioned Grantham to write new works. Grantham also collaborated with fellow composer Kent Kennan to author the textbook The Technique of Orchestration.
Well known pieces include Southern Harmony and Kentucky Harmony for band. His most famous piece is the Baron Cimetiere's Mambo.