A playmaking central midfielder who was a renowned free-kick specialist, Giunti began his career in the mid-1980s, with the Città di Castello youth system, making his professional debut in 1987, and winning the Regional Promozione divisione in 1991. He later moved to Perugia, and he was part of the team that was promoted to Serie A from Serie C1 in three years. During his first season in Serie C1 with the club, he made 20 appearances, and during the 1992–93 season, he made 28 appearances and scored 4 goals. Perugia won the 1993–94 Serie C1/B title, earning promotion to Serie B, as Giunti managed 4 goals in 24 appearances. During his first season in Serie B, he scored 7 goals in 29 appearances, and the following season, he helped Perugia gain Serie A promotion, scoring 5 goals in 25 appearances. During the 1996–97 Serie A season, he made his Serie A debut on 8 September 1996, in a home 1–0 win against Sampdoria.