A graduate of the Athenian club's youth system, Giannou was loaned to Olympiakos Volou in summer 2000 until the end of the year, and spent the rest the campaign at Halkidona FC.
Promoted to the senior Olympiacos squad on his return, he made his top-flight debut on 4 January 2002, but was mainly a reserve.
However, although by the start of 2003/04 he was still fourth-choice, a combination of Theofanis Katergiannakis's injury, Dimitrios Eleftheropoulos's departure and Juraj Bucek's loss of form saw Giannou promoted to play in the final 12 games of the league season.
He played in just one league match, when Nikopolidis was suspended, and three Greek Cup encounters as his club completed the domestic double.
He spent time on trial at Championship side Cardiff City but failed to earn a full contract.