Menzies said Iain Napier had informed the company that he wished to retire.
Mr Napier, who joined the board eight years ago, will step down after the group's annual general meeting on Friday.
He is the third high-ranking figure this year to leave the Edinburgh-based print distribution and aviation services company.
In January, former chief executive Jeremy Stafford resigned after just 15 months in the post, citing "personal reasons".
And last month, chief financial officer Paula Bell gave notice of her intention to resign in July, in order to join the board of Spirent Communications.
In a statement on Wednesday, the company said Mr Napier's wish to retire was "in line with the reduction of his other plc responsibilities".
Current non-executive group director Dermot Jenkinson will become interim chairman, following Mr Napier's departure.
He will run the process of appointing a permanent successor.
Mr Jenkinson said: "The board would like to thank Iain for his considerable contribution to the John Menzies business and board over the last eight years.
"He has marshalled us through significant periods of change over his tenure with great skill and experience. We wish him the very best for the future."
In March, John Menzies reported that contractual problems at London Gatwick Airport last year cost the company Â£6m in lost earnings.
In its annual results statement, the firm said profits were hit by increased labour costs incurred in maintaining ground handling service levels.
Overall pre-tax profit for the year was down by about 30%, at Â£18.2m.
The group's turnover for the year was flat at just under Â£2bn.
Logistics specialist John Menzies has announced that its chairman is to step down from the board later this week.