Grün started his career with Anderlecht in Belgium, where he spent eight seasons, winning the UEFA Cup in 1983, as well as three consecutive Belgian First Division titles between 1984 and 1987, among other trophies; he also reached another UEFA Cup final with the club in 1984, where they lost out to English side Tottenham, however. Grün joined Italian club Parma in 1990. During his time with the club, he established himself as one of the best defenders in Serie A, winning the Coppa Italia in 1992, and the Cup Winners' Cup in 1993 under manager Nevio Scala. Following a series of injury struggles during the 1993–94 season, however, he returned to his former club in 1994. In 1996, he returned to Italy once again, to play for Reggiana; however, he was no longer paid by the club between February and May 1997, and retired at the end of the 1996–97 season.