His decision comes after a difficult year for the organisation that saw technical director Shane Sutton quit in April after allegations of sexist and discriminatory remarks.
British Cycling has also been dragged into the controversy surrounding Team Sky's use of Therapeutic Use Exemptions (TUEs), including for former Tour de France winner Bradley Wiggins.
The organisation also said this month it was "cooperating fully" with UK Anti-Doping who were investigating allegations of "wrongdoing within cycling".
Drake, who has been involved with British Cycling for 20 years, the last eight as chief executive, confirmed that he will be leaving but not as a result of the ongoing controversies.
"Some time ago I made the decision that the Rio Games would be my last as CEO of British Cycling," Drake, who took charge in 2009, said in a statement.
"Now, following the success of our Olympic and Paralympic teams at those Games, the launch of our innovative new partnership with HSBC UK and Yorkshire's successful bid to host the 2019 Road World Championships, I believe that the end of this Olympic cycle is the natural moment for a new CEO to take the organisation forward into the Tokyo Games and beyond.
"So it has been a difficult year but my decision to move on is completely separate to that (the allegations). It's just the time is right," he added.
The British Cycling board will begin the search for a new chief executive.
"Ian has been a pivotal figure in it all," Howden said in a statement on British Cycling's website (www.britishcycling.org.uk).
"It is also testament to his commitment to the organisation that he has chosen to announce this now so that we have time to conduct a comprehensive search for his successor."
There is no suggestion of any wrongdoing by Team Sky, Wiggins or British Cycling and they have received support from UCI president Brian Cookson.
But the furore, including a claim by former Team Sky rider Jonathan Tiernan-Locke that he was offered the controversial painkiller Tramadol when competing at the road world championships four years ago, threatens to tarnish the reputation of British Cycling.
During Drake's reign Britain has grown into a cycling powerhouse and amateur participation has spiked massively.
Britain has won 20 of the 30 gold medals up for grabs in cycling at the past three Olympics.
Wiggins became Britain's first Tour de France winner in 2012 and compatriot Chris Froome has won it three times since.
(Reporting by Martyn Herman; additional reporting by Shravanth Vijayakumar; Editing by Ian Ransom/Sudipto Ganguly)
