Without further ado, let's listen to some music! For each entry I'll try to provide a genre for reference, so you can avoid specific genres you don't enjoy. This is just an indicator, so don't gut me if I accidentally get the wrong subgenre within your particular, favourite genre!
Howard Shore - Lord of the Rings soundtrack (Classical)
An obvious entry point into this list is Howard Shore's amazing soundtrack to the Peter Jackson Lord of the Rings trilogy, which, for a lot of the younger Tolkienites, might have been the backdrop to their first foray into the Tolkienverse.
"Concerning Hobbits" is a particular favourite of mine that I used to listen to a lot: Concerning Hobbits
Leonard Rosenman - The Lord of the Rings animated film soundtrack (Classical)
I've yet to watch this gem myself, but I found the soundtrack exceedingly well crafted. The movie is from 1978, and the director originally wanted to include music by Led Zeppelin, but in the end the movie was made with an original score by Leonard Rosenman. "The Company of the Ring" is one of my favourite pieces: The Company of the Ring
Clamavi de Profundis (Traditional / Classical)
A family that loves singing and music, Clamavi de Profundis is the public name of this group. They've set a lot of the poems and songs from Tolkien's books to music, and on their channel you can find music ranging from "The Song of Durin" to "The Ents' Marching Song". I'm particularly fond of the latter, because whereas the Dwarven and Elven music is pretty widely covered, this was one of the more obscure songs I've encountered. Listen to it here: The Ents' Marching Song
John Sangster - The Hobbit Suite (Jazz)
While I'm not a huge jazz-fan myself, I couldn't help but be curious about Tolkien-inspired jazz music. Sangster has made six full albums, but I found something Hobbit-related to be a good place to start. "Stompin' Under The Hill" from "The Hobbit Suite" is, in Sangster's own words, an expression of Hobbits having a good time: Stompin' Under The Hill
The Hobbitons - Songs From Middle Earth (Folk)
Dutch group the Hobbitons were given permission by the Tolkien estate to set a number of Tolkien's poems to music. That resulted in the album "Songs From Middle Earth". I really liked the good timey feel of "Old Walking Song (The Road Goes Ever On)": Old Walking Song (The Road Goes Ever On)
Za Frūmi - The Za Frūmi Saga (Folk)
This one only cropped up on my radar when I was looking for some "weird" entries to add to the list. "The Za Frūmi Saga" is basically about the adventures of a group of Orcs, and the dialogue (so not actual singing) is purely in Orcish. Interesting and strangely mesmerising!: Dushatar
Pearls Before Swine - Ring Thing (Folk Rock)
"Ring Thing" from the album "Balaclava" is, essentially, most of the Ring verse set to music: Ring Thing
Led Zeppelin - Ramble On (Rock)
Led Zeppelin have done a few songs that are in some sense inspired by Tolkien (let's not argue about Stairway). "Ramble On" appeared on a CD filled with a random selection of music that I somehow came into possession of when I was young, and references both Mordor and Gollum: Ramble On
Blind Guardian - Nightfall In Middle Earth (Heavy Metal)
Aside from the soundtrack of the Peter Jackson films, Blind Guardian were my first experience with Tolkien-inspired music. Their album "Nightfall in Middle Earth" is a retelling of the Silmarillion. One of the tracks that I keep revisiting is "Time Stands Still (At The Iron Hill)", which is about Fingolfins final showdown with Morgoth: Time Stands Still (At The Iron Hill)
Thrice - The Long Defeat (Punk)
From the album "To Be Everywhere Is To Be Nowhere", the song "The Long Defeat" shares themes with the conflicts waged by the Elves in Middle Earth. This was a new one for me, but it's got me listening to more Thrice in general: The Long Defeat
The Esgaroth Three - Who The 'Ell Is Tauriel? (Parody)
This song was made in the wake of the Hobbit trilogy, and asks the important question of just who Tauriel is, as well as making light of some of the odd decisions made throughout the three(!) movies. I found it absolutely hilarious the first time I saw the video, and I still keep revisiting it often: Who The 'Ell Is Tauriel? - NOTE: If not obvious from the title, the lyrics of the song feature some mild swearing.