2004-05: David Kveton appeared in six games for HC Vsetin in the Czech Extraliga and was the leading scorer for the club's U-20 team as a 17-year old. He scored 1 goal and was -2 in Extraliga play. Kveton scored 21 goals with 27 assists and 66 PMs in 36 regular season games and added 6 goals and 5 assists in 8 playoff games for the junior team. Kveton played for the fourth-place Czech Republic squad at the 2005 World Junior U-18 Championship. He tied for the team lead with 4 goals and had 2 assists with a +3 plus/minus and 4 PMs in seven games.
2005-06: Kveton skated in 45 games for HC Vsetin as a rookie in the Czech Extraliga. He scored 6 goals with 4 assists and 18 PMs and was -28 on a team that won six games in regulation during the season and had to avoid relegation. He also skated in seven games with second division Ostrava and five playoff games with third division Novy Jicin. Playing for the bronze medal-winning Czech Republic at the 2006 World Junior U-18 Championship, Kveton scored 2 goals with 5 assists and was +2 with 2 PMs in seven games. Kveton was selected by the New York Rangers in the fourth round (104th overall) of the 2006 NHL Draft and by the Gatineau Olympiques of the QMJHL in the first round (13th overall) of the CHL Import Draft.
2006-07: Kveton made his North American debut with Gatineau as an 18-year-old but after some early seasons struggles returned to the Czech Republic to play for HC Vsetin before later re-joining the Olympiques. In 31 games with Gatineau he scored 5 goals with 27 assists and was -2 with 17 PMs. Kveton scored 3 of his 5 goals on the power play. He had a limited role in five playoff games and was scoreless and -2 with 4 PMs. In 19 games with HC Vsetin he scored 2 goals and was -9 with 8 PMs. Kveton skated in six games for the Czech Republic at the 2007 World Junior U-20 Championship but did not have any points or penalties and was -3 in a fourth line role.
2007-08: Kveton returned to the Czech Republic to play for HC Trinec in the Extraliga and was off to a fast start before suffering a season-ending shoulder injury. In 28 games he scored 10 goals with 5 assists and was +5 with 10 PMs. Kveton skated in one game for the Czech Republic at the 2008 World Junior Championship U-20 tournament but missed the rest of the tournament due to his shoulder problem.
2008-09: Kveton was the third-leading scorer for HC Trinec in his second season with the club. Though HC Trinec finished 11th in the 14-team Czech Extraliga, the 20-year-old Kveton scored a team-leading 22 goals with 22 assists and was +4 with 20 PMs in 46 games. Kveton was selected by Atlant Mytischi in the third round (58th overall) of the 2009 KHL Draft.
2009-10: Kveton was part of a balanced HC Trinec offense bolstered by the addition of former NHLer Ladislav Kohn. Kveton skated in all 52 games for HC Trinec – which finished sixth in the Czech Extraliga and advanced to the playoff quarterfinals. Kveton scored 18 goals with 17 assists and had a team-high +17 plus/minus with 59 PMs. In five playoff games he scored 1 goal with 1 assist and had 24 PMs.
2010-11: Kveton's fourth season with HC Trinec was a championship season as the club, bolstered by the addition of former Ottawa Senator Radek Bonk, won both the regular season and playoff titles in the Czech Extraliga. Playing in all 52 games for the second straight season, Kveton scored 14 goals with 18 assists and was +11 with 26 PMs. In 18 playoff games, he scored 7 goals with 12 assists and was -3 with 8 PMs.
2011-12: Kveton was the leading scorer for HC Trinec Ocelari in his fifth season in the Czech Extraliga. Kveton scored 14 goals with 15 assists in 52 games but was minus-40 with 40 penalty minutes. With 2010-11 leading scorer playing in the KHL and one-time Dallas prospect Vojtech Polak in Switzerland, HC Trinec, the defending Extraliga champions, slipped to tenth place. The club was eliminated in the opening round of the playoffs by Zlin and Kveton had 1 assist and was plus-one with 4 penalty minutes in five playoff games.
"Good skater with acceleration, quickness and balance ... soft hands-controls puck very well... a smart player with very good vision ... a very creative offensive minded player ... has a good reach ... very good overall skill level ... a finesse type of player ... could be more agressive." - NHL Central Scouting
A finesse type of player, with speed, skill, and offensive creativity. Needs to become stronger. - ISS Scouting Report