He was runner-up in two other tournaments in Nottingham on grass to Greg Rusedski and Stuttgart Outdoor to Álex Corretja on clay.

Kučera achieved his best Grand Slam result in 1998 reaching the semi-finals of Australian Open where on his way he defeated Sergi Bruguera, Daniel Vacek, Daniel Nestor, Richard Fromberg and defending champion Pete Sampras in the quarter-finals, losing to eventual champion Petr Korda in 4 sets.

Later, Kučera was one of the contributing members on the Slovakian team which reached the final of the Davis Cup in 2005, eventually losing to Croatia 2–3.

Kučera was nicknamed the "Little Cat" because of his deceptive style of play and his fluid movement around the court resembling his coach.

[1] In 2020, Kučera was elected an MP of the National Council of Slovakia representing the Ordinary People and Independent Personalities movement, along with fellow former tennis players Ján Krošlák and Romana Tabak.