In the summer of 1967, Malibu U aired on ABC, a "variety show" with the premise of a fictional college where music stars would perform every week. It would be, for the most part, utterly forgotten (it was a "summer replacement" series with only seven episodes, although did get some good acts like The Doors) if it where not for The Ballad of Bilbo Baggins.

Star Trek had just finished its first season, and Leonard Nimoy, dressed with Spock ears, sang a song that would later come out on his 1968 album Two Sides of Leonard Nimoy. While I know people often don't stop to click links in the middle of reading a Reddit post, this one is worth it.

Tolkien did become aware of the song, and investigated as to its legal status. He eventually acceded to its existence, as long as it as it was clear he was not officially endorsing it in any way.

Two important points from this: 1.) Tolkien was protective of his universe when it came to "official" material but 2.) he was willing to let go a bizarre novelty single. In general he wasn't upset by derivative works as long as it was clear they were not part of the "primary reality" of the universe.

One derivative work came from Donald Swann, who was inspired by Tolkien's poetry while traveling and, after writing to Tolkien publisher about his new "song cycle", was invited to play it for Tolkien himself at his home. He endorsed all the songs but he noted that he had imagined Namárië rather differently, and sung for the composer what was similar to a plainchant -- the composer redid the song using that melody, which you can hear here. While Tolkien has many songs throughout his work, he was interested in the music of language, and this is the only case I know of where he actually assisted in composing a music counterpart.

Tolkien assisted in putting together the songs into a final product -- you can see a message here from Tolkien to Swann where he discusses pronunciation of Quenya words, and the book The Road Goes Ever On includes his own calligraphy and the album includes spoken material by J.R.R. Tolkien himself.

The songs definitely feel like a classical lieder but also folk, and the book specifies that the sheet music is also appropriate for folk guitar.

Now, the question was asking about slightly more modern developments in music, and unfortunately, indications are that Tolkien was not necessarily fond. While he never interacted with Led Zeppelin (that would be in the last years of his life) he did have a legendary and bizarre interaction with the Beatles.

The Beatles, like a lot of the people linked to 1960s counterculture, were fond of Lord of the Rings, and after their first two movies (A Hard Day's Night and Help!) they were encouraged by the producer Denis O'Dell to look into an adaptation featuring new music.

Peter Jackson (who has been at work on a Beatles-related project) got details from Paul McCartney about this; from a 2014 interview:

John Lennon was going to play Gollum. Paul was going to play Frodo. George Harrison was going to play Gandalf, and Ringo Starr was going to play Sam. And a lot of other people were going to play other roles. Paul was very gracious; he said: ‘It was a good (thing) we never made ours because then you wouldn't have made yours and it was great to see yours.' I said: 'It's the songs I feel badly about; you guys would have banged out a few good tunes for this. You were The Beatles, after all. It's a shame we missed out.'

The production idea was shot down by Tolkien himself. He was apparently not a fan, and one of his letters (16 July 1964) appears related:

That became hellish as soon as petrol restrictions ceased. But Headington is no paradise of peace. Sandfield Road was a cul-de-sac when I came here, but was soon opened at the bottom end, and became for a time an unofficial lorry by-pass, before Headley Way was completed. Now it is a car-park for the field of 'Oxford United' at the top end. While the actual inhabitants do all that radio, tele, dogs, scooters, buzzbikes, and cars of all sizes but the smallest, can do to produce noise from early mom to about 2 a.m. In addition in a house three doors away dwells a member of a group of young men who are evidently aiming to turn themselves into a Beatle Group. On days when it falls to his turn to have a practice session the noise is indescribable.....

It is unlikely he would have been a fan of Led Zeppelin's music, even if he was appreciative of the impulse of love for his universe.

...

Scull, C., Tolkien, J. R. R., Hammond, W. G. (2017). The J. R. R. Tolkien Companion and Guide. United Kingdom: Harper Collins.

Swann, D. (1967). The Road Goes Ever on: A Song Cycle. United States: Ballantine Books.

Tolkien, J., Carpenter, H. (2014). The Letters of J.R.R. Tolkien. United States: Houghton Mifflin Harcourt.