Scott Johnson, 33, died when the top portion of the stage fell in on him as he was setting up for the show in Toronto, Canada.
A message on the radiohead.com website said: “We have all been shattered by the loss of Scott Johnson, our friend and colleague.
“He was a lovely man, always positive, supportive and funny; a highly skilled and valued member of our great road crew.
“We will miss him very much. Our thoughts and love are with Scott’s family and all those close to him.”
Ian West, Mr Johnson’s drum teacher in Doncaster, said his former pupil toured the world with famous bands, including Radiohead and Keane.
“It was a very, very big shock. He was a great student and a great drummer. He got a lucky break and made the most of it, he never looked back.
“The list of bands he worked with was endless, his CV had every band worth their salt on it, a lot of British bands and world class bands.”
Mr West said he taught Mr Johnson for three years about 10 years ago.
“He was a fantastic guy, never down, very enthusiastic. He was very thoughtful and would do anything for you, he was spot on.”
Fellow musicians paid tribute to the drum technician on Twitter.
Jack Lawrence Brown, drummer with White Lies, tweeted: “Devastated to learn that the man killed in the Radiohead stage collapse was drum tech Scott Johnson. Scott worked with White Lies on a show earlier in the year as my drum tech.
“A very talented man and a lovely guy all round. Glad to have known him. A big loss. RIP Scott.”
Geoff Barrow, instrumentalist with Bristol band Portishead, wrote: “Shocking to hear the passing of crew member Scott Johnson of the radiohead tour - my thoughts and condolences are with everyone involved.”
Tim Burgess, lead singer in The Charlatans, wrote: “Our thoughts are with the family of Scott Johnson - the drum tech for Radiohead who tragically died yesterday.”
Radiohead, from Oxfordshire, had been due to play to a sold-out crowd at Downsview Park in Toronto but cancelled the show after the accident.
Three other people were also injured, paramedics said.
Tony Bellavance, Fire Services Platoon Chief in Toronto, said officials were alerted to a person trapped under the rubble and crews assisted in extracting the victim.
He said: “It was a crushing injury that killed the man.”
Tony Vella, a spokesman for Toronto police said crew were setting up the stage when it collapsed.
“They were setting up when the top portion collapsed on top of them. Unfortunately, four people were hurt. The remainder of the people, when they heard the stage coming down, ran from the area,” Mr Vella said.
A force spokesman tweeted that once the structure was deemed safe to work around, forensic officers would begin a joint investigation with the coroner and the Ministry of Labour.
A Foreign Office spokeswoman said: “The family has been notified and we have offered consular assistance.”