During the late 1970s and early 1980s, Cukić was the frontman of the new wave band Bulevar, releasing two albums with the band. After Bulevar disbanded in 1982, he dedicated himself to journalism. He returned to music in 1984, when he became one of the vocalists of the rock/pop rock band Bajaga i Instruktori, with which he recorded three studio albums. In 1987, he released his first solo album, and soon after left Bajaga i Instruktori to pursue solo career, which spans to the present day. In 1988, Cukić formed his backing band, Spori Ritam Band.