These jokes are typically structured in a question-and-answer session with what would purportedly be the host of the actual Armenian Radio but which would often touch topics that would be sensitive for the Communist authorities or which would otherwise be liable for censorship.
By the late 1950s, these jokes increasingly became political in nature and were actively lampooning the realities of the Soviet people, such as the lack of civil liberties, shortages, poor quality of household items, as well as satirizing Communist propaganda clichés.
Warsaw Pact countries evolved their own nuances of Radio Yerevan jokes, such as the answers of East German ones often starting with Im Prinzip ja/nein, 'In general yes/no'.
In the Soviet Union, a peculiar type of such jokes appeared that involved not the narrator but a fictional entity called the Armenian Radio.
[2] Shmelev et al. explain that the choice of the narrator was not coincidental and can be traced to "Armenian riddles" that existed since the early 20th century and featured weird questions followed by absurd answers.
— I wonder about it myself.Armenian riddles, in their turn, likely were preceded by Russian jokes popular in early 20th century that featured an Armenian, or a Georgian, answering questions (that were often asked by the Armenian/Georgian himself) in a silly way.
[1] Emil Draitser generally concurs with the assessment, further adding that the Armenian riddles saw a revival in the immediate post-World War II years.
At the same time, Communist propaganda, poor quality of life and (in Soviet satellite states) de facto lack of sovereignty made for ripe targets for satire.
[1] For a political example, Radio Yerevan defined a string quartet as the "Moscow's symphonic orchestra as it returns from a concert tour abroad", a snide remark referring to highly educated Soviet citizens migrating to the West.