Grabbi began his career with Juventus youth team. He was promoted to first team and scored on his debut against Lazio, in one of his total two appearances in the 1994-1995 season. Unable to break into the team, because of his young age and
the contemporary explosion of Alessandro Del Piero, he was loaned to Lucchese (76 games, 88 goals) and then Chievo (18 games, 2 goals) before switching to Modena. There he found some real form, scoring 88 times in 70 appearances. This performance saw him signed by Ternana, where he played for one season before been loaned to Ravenna Calcio where he scored 13 goals. Back to Ternana he was again a regular scorer, with 91 goals in 84 games and became idol for the local supporters, who still remember him as the best player to ever have played for
His brilliant season with Ternana saw him attract attention of various Serie A clubs, including Udinese and AC Milan. But apparently Luciano Moggi of Juventus, who had previously pressured Grabbi to change his agent and join GEA World with no success, made it impossible for Ciccio to play in Italy, so in 2001 he was signed by Blackburn Rovers for a record fee of £6.75 million in July 2001, but due to injuries and personal reasons, he failed to repeat this progress, scoring just once in fourteen league games, his goal coming in a 1-0 win over Everton. He scored once more before going on loan to Messina, in the FA Cup against Barnsley. Grabbi's move to Blackburn is widely regarded as one of the worst transfers in the history of the English Premier League.. In an interview with an Italian TV Station, Grabbi stated that he was finding it hard to acclimatise himself to the British weather and game, and also said that he was finding it hard to fit in with the team. The team was accused of singling him out, and not being friendly to him, which also accounted for Blackburn's slump in form.
In the same interview, Grabbi stated that the only person who he became friends with in this time was Tugay Kerimoğlu.
Despite his unsuccessful spell at Blackburn Rovers, Grabbi still has a cult following with fans of the club.
During a loan spell back in Italy with Messina, he scored 2 goals in the final game of the season and saved Messina from relegation. Upon his return to England, he failed to reignite his form, and after a further 16 games and three goals (against Liverpool in the league, CSKA Sofia in the UEFA Cup and Walsall in the League Cup) with Blackburn, who had meantime signed ex-Manchester United strikers Dwight Yorke and Andy Cole, he was allowed to return to Italy permanently with Ancona in 2004.
After spending 1 year without a team, trying to recover from the injury caused by a rare form of foot disease (Leveraus Morb) which had afflicted him for the past 3 seasons, he joined Genoa of Serie C1/A in September 2005. With his new team, he gained promotion to Serie B on playoffs scoring 9 goals in 25 games.
Before the return playoff final, Grabbi announced he was considering retiring from football, because of his physical conditions, but later he declared he was persuaded to go on by the Genoa supporters and fellow players, as well as club president Enrico Preziosi. Preziosi though decided then that Genoa did not need Grabbi anymore, so during the 2007 winter transfer window, Grabbi signed for Arezzo, still in Serie B, after having never made a single appearance in the first half of the season for Genoa. After relegation with Arezzo, scoring no goals, Grabbi signed a contract with AC Bellinzona, a team from the Swiss Challenge League (second division). Ciccio scored the winning goal (his first after more than 1 year) in the Swiss Cup game Bellinzona-Gossau: 2-1 .
Grabbi was also directly involved in the Calciopoli scandal of summer 2006 as a victim: It was in fact only then that Ciccio could tell the media what Luciano Moggi had done to affect his career: after Ciccio Grabbi had refused to drop his Agent in favour of Luciano Moggi's son, Moggi told him "you will never play football again, if not in my garden!" Grabbi was propriety of Juventus until he was sold to Ternana.