Barinka was drafted 59th overall in the 2003 NHL Entry Draft by the Chicago Blackhawks. Barinka played 2 seasons with the Blackhawks before being traded in a three-way deal to the Ottawa Senators for Martin Havlát. Barinka played for the farm team Binghamton Senators for 1 season before returning to the Czech Republic to play for the HC Vítkovice of the Czech Extraliga. After playing 5 seasons with HC Vítkovice, Barinka left to play with Lokomotiv Yaroslavl of the Kontinental Hockey League. He left to play for HC Fribourg-Gottéron of the National League A. He played 1 season with them before returning to the Czech Extraliga to play for HC Vítkovice again.