Mr Giordano will be replaced by David John, a director at the services company Inchcape, while Mr Dyer will be succeeded by Danny Rosenkranz, one of BOC's divisional directors.
In a short statement, Mr Giordano, sometimes called the "first fat cat" because of his pounds 1m salary in the 1980s, said it was time for him to move on.
His other business interests include being chairman of British Gas, where he has been a vociferous defender of the salaries paid to directors.
BOC, sometimes criticised for not putting a clear succession structure in place, said the joint announcement was intended to avoid confusion as both men came close to the end of their contracts. The new executives take over in January at BOC's annual general meeting.
Mr Giordano, 62, said: "After 15 years of service to BOC, it is time for me to move on. This change will put in place the leaders who will carry the BOC group into the next century."
Mr John is an executive director at Inchcape with responsibility for the group's world-wide Toyota Motors interests and chairman of Inchcape Berhad, the group's publicly quoted subsidiary in Singapore. He has been a non-executive director at BOC since July 1993. Trevor Taylor, currently chief executive of Inchcape subsidiary Toyota GB, will replace him.