Robin Washington, chief financial officer of biotech firm Gilead Sciences Inc, will join the board.
The changes follow scandals involving Google's handling of sexual misconduct cases and other situations that have led employees and shareholders to question the board's oversight.
Schmidt, who was instrumental in Google's growth for a decade, will remain a technical advisor to Alphabet. He had stepped down as chairman about a year ago.
"After 18 years of board (meetings), I'm following coach Bill Campbell's legacy & helping the next generation of talent to serve," he said in a series of tweets, referring to a late tech executive who is the subject of "Trillion Dollar Coach," a new book which Schmidt co-authored.
He said he would continue to serve on government technology advisory boards and teach.
Greene did not immediately respond to a request for comment. Google hired Oracle Corp veteran Thomas Kurian to lead its cloud computing unit late last year.
An employee involved in managing worker demands did not immediately respond to a request for comment.
Washington has been an executive at Gilead for about 11 years and worked at business software companies before that, Alphabet said.
Thousands of Google employees in recent months have urged the company to make corporate governance changes after it acknowledged mistakes in its handling of sexual misconduct allegations and other employee grievances. Multiple shareholder lawsuits against the board and executives stemming from those cases have also called for changes to the board and company rules.
(Reporting by Uday Sampath in Bengaluru and Paresh Dave in San Francisco; Editing by Bernard Orr and Richard Chang)
By Uday Sampath Kumar and Paresh Dave
