Kurz, who played as a defender, started playing his first football at SV Sillenbuch, a small club in his native Stuttgart, and then for VfL Sindelfingen. At age 20, he had his breakthtrough into professional football, when he signed a contract for his local Bundesliga side VfB Stuttgart in the summer of 1989. After one year, where he – with the exception of one cap – only played for VfB's second team, he was transferred to Franconian side 1. FC Nuremberg, in 1990; there he was more successful, earning 108 caps in four seasons. When the Club was relegated to the 2. Bundesliga after the 1993–94 season, Kurz took up an offer by Westphalian club BV Borussia Dortmund. The BVB won the title of German champion in the following season, with Kurz playing four times. At BVB rivals Schalke 04, where he subsequently played from 1995 to 1998, he earned 58 caps.
With Schalke he won his biggest title, the 1996–97 UEFA Cup, under Dutch coach Huub Stevens. In 1998, he transferred to the south of Germany again, joining TSV 1860 München. He played 128 times for "die Löwen" (the Lions), until 2004, when the Bavarian side was relegated to the 2. Bundesliga.
2004–05 was Kurz' last season as a player. He spent it at Swabian side SC Pfullendorf, a Regionalliga Süd (then third division) team (11 caps). Shortly after joining Pfullendorf, he was promoted to player-coach; in 2005, he retired as a player, but stayed at Pfullendorf as head coach.
After managing SC Pfullendorf, Kurz returned to 1860 München as a manager for the second team (Regionalliga Süd). In March 2007, he was promoted to head coach of the first team. 1860 München sacked Kurz in February 2009. Kurz joined 1. FC Kaiserslautern in June 2009. Kurz won promotion to the bundesliga for the club in 2010. The subsequent 2010–11 season was changeable and most of the time, FCK was threatened by relegation, but eventually the "Red Devils" finished a sensational seventh in the German top flight - their best finish in the bundesliga since 1998-99, when they finished fifth. 2011–12 was much worse, with Kaiserslautern occupying a relegation berth virtually all season. Therefore, Kaiserslautern parted ways with coach Kurz on 20 March 2012.
On 18 December 2012, it was announced by Bundesliga side TSG 1899 Hoffenheim that Kurz would be the new head coach for the second half of the 2012–13 season, when Hoffenheim faced the threat of relegation. Hoffenheim weren't able to emerge from the drop zone and decided to fire Kurz on 2 April 2013. Then Kurz was manager of Ingolstadt 04 between 9 June 2013 and 30 September 2013.