I only realized it in the last couple of minutes but I believe my first encounter with the tradition of radio taiso (radio calisthenics) was during my days at the Toronto Japanese Language School. It was kinda fun jumping and stretching about for 5 minutes on a Saturday morning in the schoolyard. Then there was the JET Programme when I joined the rest of the school twice a week in the mornings outside to do the exercises while the happy piano and the principal were leading the way. The English schools that I worked at from the mid-90s onwards didn't have that tradition but the other corporations did.
Hollywood had at least a couple of instances where the morning exercises of Japan were shown on celluloid. Although I only caught a smattering of episodes of the first season of the sci-fi show "Heroes", there was that one scene in Episode 1 where Hiro Nakamura was doing his calisthenics with the rest of his co-workers. And I even found this scene from the old "Gung Ho" with Michael Keaton (about a Japanese corporation taking over a US car factory) when he was still largely known as a comic actor (still waiting for him to do "Beetlejuice 2"). The workers didn't seem too gung-ho to warm up.
Ironically, though, according to what I found out from the J-Wiki and Wikipedia articles on morning calisthenics was that radio taiso was inspired from the 15-minute radio broadcasts of exercises performed in the United States in the 1920s.
Although radio exercise broadcasts had been attempted in Japan slightly earlier but without musical accompaniment, it was from November 1928 that the first radio taiso known as Radio Taiso One with music was broadcast. The exercises were done to a song titled "Kawaii Kashu"（可愛い歌手...Cute Singer）composed by Naoaki Fukui（福井直秋）with lyrics added later by Toshiko Takeuchi（武内俊子）. You can listen to the Miku Hatsune（初音ミク）cover above with a broadcast from those days.
This first version lasted from 1928 to April 1946 although the broadcasts were halted for a week between August 15th and 22nd 1945. There was a second version with a short lifespan between April 1946 and August 1947 with the music provided by classical composer Tadashi Hattori（服部正）.
The third version of Radio Taiso One and the version that has continued to the present day started from May 1951 with the same composer Hattori. Almost feeling like standing up and doing some of them again...no, not really.
In parallel with Radio Taiso One, there has also been a Radio Taiso Two with different incarnations over the decades beginning in 1932. The current Two began in June 1952 that was composed by Ikuma Dan（團伊玖磨）. My time at the Tsukiyono junior high schools on JET involved me doing both Radio Taiso One and Two.
There was even a Radio Taiso Three that had two versions in 1939 and 1946, both by Kunihiko Hashimoto（橋本國彦）but the second version ended in 1947. In 1999, the 5-minute program "Minna no Taiso"（みんなの体操...Exercise for Everyone）was created by NHK and my family sees it almost everyday at 12:55 pm EST via TV Japan. We politely watch only. According to J-Wiki, this has been dubbed "Radio Taiso Three" although I'm not sure whether the originating network ever officially designated it as such. The music was composed by Toshihiko Sahashi（佐橋俊彦）who has provided music for TV, movie and anime soundtracks.
So, for those folks who have just joined JET (I think the Orientation sessions have just opened in Tokyo this week): first off, welcome to the Programme, and second, look forward to exercising with the teachers and students!
Of course, you can take a look at how this famous technopop band took an affectionate poke at the cultural custom.