Cross is to step into the role within the next few months. He is currently managing partner at Portas, which was previously called Yellowdoor.
Dickinson, who joined John Lewis in 2000, is leaving to take up consultancy work.
Cross will report to John Lewis MD Andy Street, who said: ‘Peter inherits an award-winning and highly effective comms team that has been developed under Helen's excellent leadership.
'He brings a unique mix of skills and unparalleled experience to John Lewis. It is extremely rare to find a comms specialist with such a broad understanding of the UK consumer and the revolution under way on the British high street. We are looking forward to him joining us immensely.’
Cross said: ‘To be asked to lead the comms challenge at John Lewis is a great privilege. I’ve watched the extraordinary rise and rise of Britain’s favourite retailer during recent years and am looking forward to helping Andy achieve the brand’s bullish ambitions over the next few years.’
At Portas, Cross worked alongside Mary Portas and has worked with brands including Westfield, Louis Vuitton, Clarks and Mercedes-Benz.