- Bats Left, Throws Right
In the 2001 European Championship, Veselý was 1 for 16 with a walk and two strikeouts; he also tossed a scoreless inning, walking one and fanning two. For the first season for which Extraliga stats are available, Veselý hit .354/.454/.381 for Draci Brno in 2003 and stole 17 bases in 19 tries. He tied for 4th in the league in hits (40), was 10th in average, was 9th in slugging, tied Jan Toman for 5th in walks (21), was 8th in OBP, 10th in RBI (22) and fourth in steals. During the 2003 European Championship, Martin was 4 for 21 with four runs but drew 7 walks. He tied for fourth in the tournament in walks. He also stole 3 bases in 3 tries, tying Javier Zabalza, Chris Demetral, Darren Heath, Raily Legito and Gaspard Fessy for 7th in swipes.
Veselý hit .356/.438/.462 and scored 32 runs in 28 games in 2004. He tied for 5th in hits (37), tied for 8th in total bases (48, even with Aleš Navrátil and Jakub Malik) and tied for the most runs. In 2005, Martin batted .411/.515/.496 for Draci Brno with 28 walks, 33 RBI, 45 runs and 20 steals (in 23 tries). His 53 hits were second to Malik. He was third in average (behind Jakub Hajtmar and Malik), third in total bases (64), was 5th in walks, placed 2nd in OBP (behind Tomáš Polanský), was 4th in RBI, stole the most bases and scored the most runs. He hit .524/.545/.810 in the 2005 European Cup, which he led in slugging (.096 over former AAA player Evert-Jan 't Hoen), doubles (4), total bases (17, 5 more than Gabriele Ermini), triples (1) and hits (11, tied with Ermini). He was 4th in OBP (between Ruslan Nabiev and Octavio Medina) and 3rd in average (after Ermini and 't Hoen). He batted .240/.375/.360 with 5 runs in the 2005 European Championship and was 3-for-3 in steals. Only Michal Kočí had more steals for the Czechs and only Hajtmar scored more runs.
In 2006, Veselý's batting line was .448/.543/.629 with 37 runs in 32 games and 10 steals in 12 tries. He was second in hits (52, trailing Hajtmar), second in average, led in total bases (73, one ahead of Pavel Budský), was third in slugging, tied for second in walks (23, even with Kočí and Jan Jablonka), tied for third in steals, was second in OBP (by .001) and was second in runs (one behind Budský).
Veselý hit .371/.519/.546 in 2007 with 29 runs and 30 walks in 28 games. He was 2-0 with no runs allowed in 4 1/3 innings as a pitcher as well. He tied for third in hits (36), was 4th in average, 3rd in total bases (53), 4th in slugging, 2nd in walks (one behind Jablonka), 2nd in OBP (trailing Jablonka), tied for third in RBI and led in runs.