In many Vietnam war films, Credence Clearwater Revivals Fortunate Son can often be heard, but was the song popular among the soldiers? Was music like that allowed to be played on helicopters?

OK SO I'm going to share my research process on this at the end because I found it quite funny.

The answer to this is yes, conditional upon the musical tastes of the troops obviously. I wrote this post which /u/ForExes kindly dug up(and called excellent! I'm blushing!) awhile ago and I think that answers the popularity aspect of this music. I would also encourage you to read all the comments replying to mine in that answer because the users tidied up a few of my scattered thoughts and incorrect points.

The second part of your question is what I'll address here. Again, the answer is yes, and not just allowed, but encouraged.

Dave Billingsly, an American medic who arrived in Vietnam in 1970 recounts his first day in Vietnam:

I saw this little club there called Alice’s Restaurant,” Billingsly reported. “Then, a helicopter flew over and it had a peace sign painted on it. It also had a speaker hanging out of it, and they were playing ‘Sgt. Pepper’s Lonely Hearts Club Band.’ So, I went into Alice’s Restaurant and everywhere I looked there were dudes with peace signs around their necks, beads, headbands. And I’m thinking to myself, ‘Where the fuck am I?’

This account shows that among the troops, longing for peace was a common sentiment(unless he walked into the pacifist hotbed of the American Army on day 1). This means that many of the protest songs, like Fortunate Son, would resonate with an army of men who didn't particularly want to be there. More directly to your second question, it shows that music was played from the helicopters.

A similar account says:

We’re flying at seven or eight thousand feet listening to AFVN radio—‘From the Delta to the DMZ’— and ‘Like a Rolling Stone’ comes on. The rim shot and the explosion of the music. I could remember it from 1965 when we used to just turn it up and stop everything to listen to it, but flying over Vietnam, in the cockpit of an airplane, being totally stoned, out of bullet range, looking down at the war, and listening to the music, nothing like it.

Again, music is a common theme here(along with pot). I guess the new question is why was music allowed on the choppers? Surely this would be distracting or something? The answer actually links back to your first question about popularity.

In the dying years of the war, American troop numbers were being scaled back in preparation for an exit. This led to more men resisting fighting(disobeying orders, fragging officers etc.) and in general, there was the idea that control of the troops might have been lost. The American response to this was to raise morale with material things. Classic capitalists.

Meredith Lair in her book Armed with Abundance: Consumerism and Soldiering in the Vietnam War writes that the military hierarchy made sure that music was available to the troops at all times. By 1969 she says, a third of troops listened to music for 5 hours a day. About a third of your waking time in a standard 8-16 hour sleep schedule, but I assume soldiers would have less sleep time. 99% of personnel had a radio by 1970, and the American Forces Vietnam Network(who I think is GOOOOD MOOOOOOOOOORNING VIETNAAAAAAAAM?) were broadcasting 24/7.

So, music was seen as a key weapon in the war against declining troop morale, to the point where the military even organised Woodstock tribute concerts for the troops, and encouraged them to listen to music. I don't know how loud you'd have to play a song to hear it over the rotors, but by the two accounts above, that's what they did.

EDIT: My research process! My first move was to Google "did vietnam helicopters play music" because that seemed like it might lead to some obscure article.. Instead it led me to this absolute gold mine of Yahoo Answers, where good ole patriots told me that the only thing that helicopters played was machine guns and rockets! They also provided several illuminating answers about Hollywood BS, and one guy suggesting that playing music might give you away to the enemy, as opposed to the sound of your rockets?

Anyway, I then found a chapter of this book which is where I got the quotes, the references to Meredith Lair's book, and the general information. I assume that book is available in libraries, but it's quite a reasonably priced text and looks like an interesting read. If you want to buy it for a poor student, please PM me ;) Hope this has been an interesting answer!

I had actually done some research myself into the subject of music in Vietnam regarding what was allowed and what wasn't and I found this excellent piece from 2000 by Chris Sabis as his thesis at University of Rochester regarding music in Vietnam. He goes into detail about the effect of music during the war, as well as how AFVN (Armed Forces Vietnam Network) selected and played music.

Basically the idea was that AFVN was regulated by what they could play, but the regulations were rather interesting. Despite songs being restricted for drug references, anti war themes, and questioning of leadership, there were times where unauthorized songs would be played by DJs. This didn't amount to much punishment to such acts. Also in there was a mention of DJs being allowed to play the protest song "Ohio" citing that this may have been more of a mockery of the incident due to soldier's resentment of college students at the time, including DJs making subtle jokes on the matter. There were also pirate and illegal radio stations in country at the time as well.

As for blaring "fortunate son" out of helicopters, there were instances of loudspeaker-laden UH-1s and such using music and other recordings for PSYOPS missions as well as the Chieu Hoi program, with some information on this site

I doubt fortunate son would be one of those songs due the nature of it