Rodriguez was born in Cuba and moved with his family to Jamaica at an early age.He grew up in Kingston, Jamaica, 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 became closely musically related to rasta drummer Count Ossie. In 1961, Rodriguez moved to the UK and started to play in reggae bands there.In 1976, he recorded the album, Man from Wareika under contract with Island Records.In the late 1970s, with the arrival of the 2 Tone genre, he played with ska revival bands such as The Specials. One of his most notable performances was on The Specials' song "A Message to You, Rudy".
Rodriguez formed the group Rico and the Rudies to create the albums Blow Your Horn and Brixton Cat.
In 1995, Island Records released the album Roots to the Bone, an updated version of Rodriguez's earlier work, Man from Wareika.
Since 1996, among other engagements he has played with Jools Holland's Rhythm and Blues Orchestra and also performs at various Skafestivals throughout Europe with his own band. He performed with Jools Holland and his R & B Orchestra for a number of years, until retiring from the band in 2012.
He was awarded an MBE (Member of the Order of the British Empire) at Buckingham Palace on 12 July 2007, 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.
On the 4th September 2015, following a short spell in hospital, Rico died at the age of 81.