Welsch began his professional career as a teenager with BHC SKP Pardubice of the Czech Republic, where he played until 1998. He signed with Sparta Prague (Czech Republic) for the 1998-99 season, and then with Olimpija Ljubljana (Slovenia), where he played until 2002. He was named Mattoni League Most Valuable Player following the 2001-02 campaign.
Welsch was drafted by the Philadelphia 76ers with the 16th selection of the 2002 NBA Draft, but was immediately traded to the Golden State Warriors. He spent one season with the Warriors (1.6 ppg, 0.8 rpg, 0.7 apg in 37 games) in 2002-03 before he was traded to the Dallas Mavericks, along with Antawn Jamison, Chris Mills and Danny Fortson, in exchange for Evan Eschmeyer, Nick Van Exel, Avery Johnson, Popeye Jones and Antoine Rigaudeau on August 18 2003. Without ever playing a game for them, Dallas traded Welsch, Mills, Raef LaFrentz and a lottery-protected pick in 2004, on October 20 2003 to the Boston Celtics in exchange for Antoine Walker and Tony Delk.
Welsch became a solid contributor for the Celtics (8.5 ppg, 3.2 rpg, 2.0 apg in 136 games), starting 100 games in 2003-04 and '04-05. Despite his improvement, Welsch was traded to the Cleveland Cavaliers on 24 February 2005, where he spent the remainder of the 2004-05 season (2.9 ppg, 1.8 rpg, 1.2 apg in 16 games).
Regular Season Stats
|2002-03||23||Golden St. Warriors||37||234||61||1.6||19||75||.253||22||29||.759||1||4||.250||28||0.8||27||0.7||8||2||19|
|3 Season Totals||189||3730||1268||6.7||430||1058||.406||304||404||.752||104||295||.353||491||2.6||313||1.7||153||14||221|