The singer pulled out of a promotional appearance on the BBC Radio 1 Breakfast Show in support of her seventh studio album Unapologetic.
In response to the cancellation, Grimshaw told listeners that he will not play any of Rihanna's new album until she reschedules the interview.
"I'm gonna have a tantrum because it's my favourite popstar," he said on air. "We're not even mentioning her. We're not gonna promote it now.
"We were kinda forced into having Rihanna as our Record of the Week because we were going to do a big interview with her but once it was cancelled we changed it."
He added: "She ruined my week. Nice one Rihanna."
Producer Matt Fincham quipped: "We don't want to give her publicity until she appears on the show."
Rihanna is currently on course to claim her fourth UK number one album with Unapologetic on Sunday (November 25).
Watch the music video for Rihanna's 'Diamonds' below: