Giraldi, hired in October, joined the promoted club after they had made more than 20 signings in the summer ahead of their long-awaited return to the top flight.
With the Italian on board - formerly Chief Scout and Technical Director at Watford for more than eight years - Forest added another seven signings in the January window, but the results were mixed to say the least.
He left the team in the relegation zone and winless in their last nine games, with an immediate return to the Championship looking likely.
"Filippo would like to place on record his thanks to the club and wishes Nottingham Forest best of luck for the future," the club said, adding that he left "amicably".
Forest owner Evangelos Marinakis released a statement last week to dispel rumours of possibly sacking coach Steve Cooper (43), saying the club were in a difficult place but the manager's position remained unchanged.
They're the only club in the bottom five of the league that haven't yet opted to change their manager this season.
Next up for the side, 18th in the Premier League on 27 points from 30 games, is a home match against Manchester United on Sunday.
