The firm has started the search to replace Bickerstaff who will remain in his current position until the end of November.
He steps down at a challenging time for Costain as it battles to restore its share price from around 40p to its level of around £4 two years ago.
Last month Costain announced a £92m first half loss after taking big hits on the A465 Heads of the Valleys road contract and the Peterborough & Huntingdon gas station job.
Earlier this year Dubai-based contractor ASGC injected £100m into the business taking a 15% stake..
Dr Paul Golby CBE, chair, said: “Tony has been with Costain for 14 years and I would like to thank him for his significant contribution and personal commitment to the Group during this time.
“He has been a valued member of the board and we wish him every success in his next career move”.
Bickerstaff said: “I have enjoyed my time at Costain and I’m proud to have worked with so many talented people. I look forward to my next challenge and wish the board and my colleagues the very best for the future.”