Rodriguez was born in Cuba and grew up in Jamaica, before moving to England in the 1960s, according to the Jamaica Observer.
As well as playing with The Specials, Rodriguez performed as a solo artist, recording albums including Man from Wareika, Blow Your Horn and Brixton Cat with his band Rico and the Rudies.
He worked with many different musicians and producers during a lengthy career, including Prince Buster, Karl Pitterson and Jools Holland in his Rhythm and Blues Orchestra.
He was made an MBE in 2007 for services to music.
Among those paying tribute has been Elvis Costello, who tweeted: What a moment it was to hear him play on "Rudy".
REST IN PEACE RODRIGUE
