Elliott was born and raised in Baltimore, Maryland. He originally planned to become a studio musician and studied organ as part of this goal. After studies at the preparatory division of the Peabody Conservatory and the Catholic University of America, he received a bachelor's degree from the Curtis Institute of Music while also serving as an assistant organist for the Wanamaker Organ. It was while he was a student at Curtis that Elliott joined the LDS Church. He served a mission for the LDS Church in Argentina from 1981 to 1983.