Henryk Kukier had a lengthy career as an amateur flyweight boxer that lasted from 1948 through 1961. During that time, he represented Poland at three editions of the Olympic Games, being eliminated in round one in 1952 and round two in 1956 and 1960. He was more successful at the European Championships, taking gold in 1953 by defeating František Majdloch of Czechoslovakia and bronze in 1955, in addition to competing in 1957 and 1959 without reaching the podium. Domestically, he won six national championships, from 1953 through 1957 and in 1960, and worked as a coach and trainer after his retirement.