Born in Naples, Campania, Grieco started his career at Ligurian side Geona. In late 1990s (in the 1999–2000 season or earlier) he was sold to Parma in a co-ownership deal but was loaned back to Genoa. Grieco made his debut on 12 September 1999, coming on as a substitute for Alessandro Manetti in the 82nd minute against Pescara. Genoa lost the match 1–2.