Murray studied at Butler University and the Oberlin College Conservatory of Music, before private study with Marcel Dupré in Paris. (He was the last important student of Dupré, of whom he would later write a biography.) During the 1968-69 performance season, Murray performed the complete organ works of Bach in a series of twelve recitals in Cleveland. He later commemorated the 150th birthday of César Franck by playing that composer's complete organ works in 1972. He repeated the cycle for the hundredth anniversary of Franck's death (1990, at the Cathedral of St. John the Divine in New York City. Murray's performances have included nearly every major city in North America, with numerous appearances as a soloist with major orchestras. Critics hailed Murray's rare combination of technique, thoughtfulness, and musical feeling.
Murray has written many articles and has published four books. His Marcel Dupré: The Work of a Master Organist (Northeastern University Press, ISBN 0-930350-65-0) is in its third printing. He has also written French Masters of the Organ (Yale University Press, ISBN 0-300-07291-0) and served as editor for A Jacques Barzun Reader (HarperCollins, ISBN 0-06-093542-1). He has made many recordings on the Telarc label, featuring the works of Bach, Franck, Saint-Saëns, and others.
Murray is now retired from concert performing and recording, and is the assistant organist at St. Mark's Episcopal Church in Columbus, Ohio. He has been awarded an honorary doctorate by Ohio State University.