In 1963, Kenney began his radio career at the age of 15 as a disc jockey at WIRL in Peoria. After WIRL, he worked at WOWO, Ft. Wayne, Indiana; WKYC (AM, now WTAM), Cleveland (1970–1973);WHN, New York City WYNY, New York City; WJJD (now WYLL), Chicago; and WKHK (now WLTW), New York City until 1972.
Kenney was part of the regular cast on the Imus in the Morning radio show from 1973 to 2007, where he recorded impersonations of dozens of characters, including General George Patton, Andy Rooney and Ross Perot. Also in 1973, Kenney joined 1050 WHN (now WEPN), a country music station in New York City. He originally announced the afternoon drive-time, before taking over the prestigious morning drive-time in 1974. He stayed at WHN until the fall of 1979. His show was a hit, not only for the music, but also for his comical characters who "joined" him while performing his deejay duties. He was the host of the New York edition of the television show Bowling for Dollars on WOR-TV (now WWOR-TV) from 1976 to 1979.
He is also known for his voice work as Lion-O on the 1980s Rankin-Bass cartoon ThunderCats, Clarence Thomas, and Karate Kat, a martial arts blackbelt cat featured as part of The Comic Strip. He was also the voice of Bluegrass in SilverHawks. Kenney also did voice work for several breakfast cereal characters, including Count Chocula and Sonny the Cuckoo Bird. In recent years, he has reprised this role for humorous ThunderCats references on the animated series Family Guy. Warner Bros announced that a new Thundercats animated series is in the works, set to air in 2011 on Cartoon Network. Kenney will return to the series, but will be voicing Claudus, Lion-O's and Tygra's Father.
Kenney provided voice-overs for The State, the 1990s sketch comedy cult classic which featured his daughter, Kerri Kenney. He was also the announcer for VH1's Best Week Ever during its run from 2004–2009, and provides introductions for Westwood One's radio coverage of Monday Night Football and various other commercial work. He was the announcer for "102.7 The Beat" in the video game Grand Theft Auto IV.