Rodriguez was born in Havana, Cuba, and at an early age moved with his family to Jamaica.He grew up there in Kingston, and was taught to play the trombone by his slightly older schoolmate Don Drummond at the Alpha Boys School. In the 1950s, Rodriguez became a Rastafarian and was closely associated musically to the rasta drummer Count Ossie.
In 1961 Rodriguez moved to the UK, where he joined live bands such as Georgie Fame's Blue Flames and started to play in reggae bands. Rodriguez also began recording with his own band, Rico's All Stars, and later formed the group Rico and the Rudies, recording the 1969 albums Blow Your Horn and Brixton Cat. In 1976 he recorded the album Man from Wareika under a contract with Island Records. In the late 1970s, he recorded a song called Offshore Banking Business with The Members and with the arrival of the 2 Tone genre, he played with ska revival bands such as The Specials including their single "A Message to You, Rudy".
In 1982, he returned to Jamaica to retire from performing professionally; however, in 1987 he returned to tour with the Heart Beat Band, and between 1992 and 1995 he would also play with Jazz Jamaica, as well as with Linton Kwesi Johnson during this era.
In 1995 Island Records released the album Roots to the Bone, an updated version of Rodriguez's earlier work Man from Wareika. From 1996, among other engagements, he played with Jools Holland's Rhythm and Blues Orchestra and also performed at various ska festivals throughout Europe with his own band. He retired from performing with Jools Holland in 2012.
He was appointed a Member of the Order of the British Empire (MBE) in the 2007 New Year Honours, for services to music. In October 2012 he was awarded the Silver Musgrave Medal by the Institute of Jamaica in recognition of his contribution to Jamaican music.
