The mass resignation came two weeks ahead of the their gala awards ceremony and just days after more than 200 leading film actors, producers and directors demanded "profound reform" of the industry.
"To honour those men and women who made cinema happen in 2019, to find calm and ensure that the festival of film remains just that, a festival, the board... has unanimously decided to resign," the academy said in a statement.
"This collective decision will allow complete renewal of the board," it added.
The move, which has been welcomed by feminist groups, follows weeks of controversy over the academy's attitude to women and its lack of transparency.
In mid-January there was anger after the academy refused to let two young actresses invite their mentors, film-makers Virginie Despentes and Claire Denis, to a dinner organised to honour the up-and-coming actors. Despentes and Denis are outspoken feminists.
The academy came under fire again at the end of January after Roman Polanski's film "An Officer and a Spy" topped the list of nominations for this year's Cesar awards, due to be handed out on February 28.
Polanski is already persona non grata in Hollywood over the 1978 statutory rape of a 13-year-old girl. More recently he has denied new rape allegations from photographer Valentine Monnier, who says he raped her at his Swiss chalet in 1975.
France's equality minister, women's groups and some film critics condemned the inclusion of Polanski's film on the Cesars' shortlist. But president of the academy Alain Terzian said it could not be expected to take "moral positions" when evaluating films.
The open letter from leading film industry figures, published late Monday in the French daily Le Monde, called for reform of the academy's statutes, "which have not changed for a very long time".
It protested that its 4,700 members had no say on how the academy, or even its awards ceremony, was run. It attacked its lack of transparency and the under-representation of women.
Signatories included many of French cinema's brightest and best: actors Mathieu Almaric and Elodie Bouchez; directors Bertrand Tavernier and Jacques Audiard; and producer Michel Hazanavicius, who picked up the 2011 Best Picture Oscar for The Artist.
In response, Terzian announced reforms were on the way, including moves to remedy the under-representation of women.
In the end however, the growing pressure proved too much.
Thursday night's statement announcing the unanimous decision of the board to resign said this would allow for a complete overhaul of the academy at a general assembly to be held after the February 28 awards ceremony.
'When we mobilise, things happen!'
French feminist groups had already demonstrated outside cinemas showing Polanski's film. When it topped the list of nominations, they called for those with votes at the academy to snub the film.
And in an open letter to the press on Tuesday, they called for protests outside the award ceremony itself, in Paris's Salle Pleyel auditorium.
On Thursday night, they were celebrating news of the board's resignation.
"When we mobilise, things happen!", feminist collective Nous Toutes (All Of Us) wrote on Twitter.
Another group, Osez Le Feminisme (Dare Feminism), said: "Imagine what's next. A new voting panel without male self-confidence, opacity and sexism. Will we finally stop applauding rapists and paedophiles on the run?"
But feminist groups are still calling for the protests outside the Cesars ceremony to be maintained.