Urbaniak started his music education during high school in Łódź, and continued from 1961 in Warsaw in the violin class of Tadeusz Wroński. Learning to play on the saxophone alone, he first played in a Dixieland band, and later with Zbigniew Namysłowski and the "Jazz Rockers", with whom he performed during the Jazz Jamboree festival in 1961. After this, he was invited to play with Andrzej Trzaskowski, and toured the USA in 1962 with his band "The Wreckers", playing at festivals and clubs in Newport, San Francisco, Chicago, Washington, and New York City.
Urbaniak has invited and has been invited by many other well known jazz stars, including Lenny White, Wayne Shorter, Marcus Miller, Joe Zawinul, Ron Carter, Kenny Barron, Buster Williams and Quincy Jones. In 1985, he was invited to play during the recording of Tutu with the father of fusion, Miles Davis. Davis is reported to have said on this occasion: "Get me this fucking Polish fiddler, he's got the sound!"