It depends on the songs themselves - some songs were popular at the time, while others probably only gained favour with audiences later.

'Fortunate Son' reached #3 on the Billboard charts in 1969, so it was definitely popular with contemporary audiences. It was also very obvious that the song was about the war, and more specifically, the draft and the way it unfairly targeted certain members of society.

Crosby, Stills, Nash & Young released 'Ohio' in 1970 in response to the Kent State shootings where the Ohio National Guard killed four unarmed students, while attempting to shut down a protest against the Vietnam war. That song reached #14 on Billboard.

A lot of these songs were popular at the time because they emerged from the Sixties counterculture that was also heavily grounded in the anti-war movement. Songs like 'Fortunate Son' resonated with certain groups, like students, as they became more politically aware and recognised inconsistencies in American society.

The Kent State protests, which inspired the song 'Ohio', occurred in 1970, while there were massive protests occurring back in 1968 at Columbia University over the same issues - race and war. The students at Columbia protested against their University having association with a think tank related to arms manufacturers working with the DoD. As well as that, there were plans for a gymnasium to be built for the University, but it was to be built on land which was already occupied by low income housing but it was to be built on public land in Harlem, and segregated entrances for public users(mostly black) and university users(mostly white).

I bring in this extra detail to point out that what we think of as protest songs, and might see as being unpopular at the time, were actually a reflection of the mood of a large portion of the population in the years of the Vietnam war.

EDIT: fixed Kent State student sentence re: /u/tractorguy's comment below.

Op, I think tastes would have been similar, but with limitations. The people at home were protesting a war, while those away were already stuck in it. I'm not sure how they'd feel. Hopefully someone else can explore this.

EDIT 2: I got my usage of land mixed up! AS /u/Silver_kitty pointed out, the university had been buying buildings and evicting tenants too. I'd connected that with the gym first, when I believe it was a whole other issue. 1968 was a hell of a year.

'All Along The Watchtower' (according to Joel Whitburn's chart book The Billboard Book of Top 40 Hits) was a #20 single in September 1968 - it's actually the only Jimi Hendrix single to chart in the top 40 in the US.

'Fortunate Son' was a Double A-Side with 'Down On The Corner' (which meant that 'Fortunate Son' was on one side of a 7" record, and 'Down On The Corner' on the other, and that the record company would often print two versions of the cover, one promoting 'Fortunate Son', one promoting 'Down On The Corner'). 'Down On The Corner' was a #3 single, and 'Fortunate Son' was a #14 single.

Additionally, both songs came from successful albums - Hendrix's Electric Ladyland was a #1 album, and CCR's Willy And The Poor Boys was a #3 album.

So I think it's fair to say that they were actually popular during the war.

In contrast, 'What A Wonderful World' by Louis Armstrong - very closely associated with the movie Good Morning Vietnam - was not a popular hit in the US at the time; according to Whitburn it was a #116 hit in 1968.

So, some songs now associated with the Vietnam War were popular at the time, and some weren't.

EDIT: In regards to what the troops listened to, some playlists from AFVN radio in early 1969 and late 1968 are apparently archived here. This is before 'Fortunate Son' was released, but 'All Along The Watchtower' was featured prominently on a playlist from November, suggesting it was popular amongst troops. The playlists on AFVN look like they feature basically the same songs as the Billboard charts the same month.

As a related question, I'd like to know which songs were particularly popular among African-American troops of the era. It seems likely that some songs/artists had crossover appeal to multiple demographics, but as racial tensions were very much present both at home and in theater, were there any artists (especially those of color) who produced what black troops considered "their" music?

An interesting example during the war was a legendary underground radio station, Radio First Termer which was operated in Saigon by Air Force Sargent Clyde Delay, aka Dave Rabbit who played in his own words "Hard Acid Rock" associated with the counter culture. This may be an isolated example of the GI's preferences since the studio only operated for a month in 1971, but its evidence that there was an audience that wanted to hear the same tunes of the people that were protesting the war. Here is some surviving audio

I asked a question that was deleted (no idea why...), so I guess I'll repost it.

Was the music that AFVN played sanitized? Did they choose not to play protest songs?