His death meant Friday’s showing of Mack and Mabel at the Theatre Royal was cancelled.
His agent tweeted: “The talented and much loved actor @GianniniAlex died suddenly of natural causes yesterday.”
Actress Anna Marie Cseh said on Twitter it was a “real pleasure and privilege” to work with him.
Giannini, 52, also appeared in Elizabeth: The Golden Age and Miss Monday, and featured in TV shows such as The Bill and Dalziel and Pascoe.
Agents Felix de Wolfe added: “An incredible life force has left the building.”
Stephanie Sirr, chief executive of Nottingham Playhouse tweeted: “Shocking, terrible news. A lovely man with incredible zest for life. A great performer.”
The performance on Friday evening was cancelled about an hour before the curtain was due to rise.
David Bloom, of the Chichester Festival Theatre production company, said: “There has been a company bereavement and so the performance tonight has had to be cancelled.”
All remaining performances would go ahead as scheduled, the theatre said.
The theatre wrote on Facebook: “Thank you to everyone who was affected by last night’s cancellation, for your patience and understanding.”
Theatregoer Debbie Lumsden said on the same Facebook page: “Very sad news which was handled with great sensitivity by all the staff at the Theatre Royal last night.”
Fairy Powered Productions send condolences to his family and friends