Murray was initially heavily influenced by free jazz musicians such as Archie Shepp and Albert Ayler. He gradually evolved a more mainstream approach in his playing and compositions.
Murray was a founing member of the World Saxophone Quartet.
One critic dubbed Murray the Joyce Carol Oates of jazz, comparing Murray's prolific and consistently highly-regarded work to the noted novelist's.
Murray has recorded or performed with musicians such as Henry Threadgill, James Blood Ulmer, Olu Dara, Butch Morris and Steve McCall.
see also List of jazz musicians