I haven't yet watched the entire trilogy, just bits of it (read all the books) but
1. In the LOR and the Hobbit all the characters sing, at almost any opportunity. Even the Barrow Wight sings. This I think relates to the way things were before the industrial revolution and still are in many traditional societies - in the "less civilized" parts of the world - many people would sing as they worked - or if you had to travel somewhere you'd sing as you walked. As the hobbits did, they often sang as they walked (mentioned several times). I liked the song of the dwarfs at the start of the Hobbit movie - one of the few things I've liked in the movie adaptations - should have had a lot more of that.
2. I think the Tom Bombadil omission makes far more of a difference to the tone than most realize. That's because he is the only person in the entire book who the ring has no influence over. For him it is a trinket that has no power over him, and of little interest which he would soon lose or forget. I think that perspective makes a difference to the whole book, even though it is a short incident in the first book that most probably forget - still - that there is at least one person / being in Middle Earth that the ring has no hold over.
3. The emphasis on battles, and the action sequences. In the Hobbit movies this just went on and on - and so improbable, characters falling for tens of meters and then bouncing up unhurt, like sprites in a computer game. The stories are much more fun and enjoyable - and the adventures more believable because they happen in a context that is in other ways so ordinary, and because they are not over the top.
Sorry realize I haven't really answered the question as posed.
First, most of them are meant to be sung. You write a bit differently for songs than for poems to be read out loud, so I think you miss something by just reading them without music, like, think of them as sung even though they don't have written tunes.
There are a few recordings of Tolkein himself singing humorous songs from the books. So we have tunes we can use for those (though no reason not to make up new tunes for them also).
and
He didn't attempt to sing any of the moving and beautiful songs as far as I know just these humorous ones. It would be interesting to know if he had particular tunes in mind for any of the other ones.
I think also brings out the theme of an earlier world not yet touched by the shadow of Sauron, where people sung in a care free way, just naturally as part of the way they lead their lives, like the singing in Tom Bombadill's house. As Alastair Milne says in comment below (example of "Ho Ho HO"), the darkness then comes over so much more strongly because of that.
Songs all the way through the book like that, either just before the moments of fear and darkness (the Ho Ho Ho), or just after (like the song of Tom Bombadil for old man willow and barrow wight), or in the middle of it.
Even on the edge of Mordor, Sam reciting his poem about the elephant, though not sung, but it's the same kind of thing.
Often you can almost hear the song as you read it. You can try writing down the tunes - I did that, was a fun project, wrote tunes eventually for all the songs in LOR and the Hobbit. The tunes almost wrote themselves, many of them.
I'll add a couple of links to my answer.
This is a list of all the poems in LOR, many of them sung.
Poems in The Lord of the Rings
And this one talks a bit about each one.
About the Songs and Poems in the Lord of the Rings
1. In the LOR and the Hobbit all the characters sing, at almost any opportunity. Even the Barrow Wight sings. This I think relates to the way things were before the industrial revolution and still are in many traditional societies - in the "less civilized" parts of the world - many people would sing as they worked - or if you had to travel somewhere you'd sing as you walked. As the hobbits did, they often sang as they walked (mentioned several times). I liked the song of the dwarfs at the start of the Hobbit movie - one of the few things I've liked in the movie adaptations - should have had a lot more of that.
2. I think the Tom Bombadil omission makes far more of a difference to the tone than most realize. That's because he is the only person in the entire book who the ring has no influence over. For him it is a trinket that has no power over him, and of little interest which he would soon lose or forget. I think that perspective makes a difference to the whole book, even though it is a short incident in the first book that most probably forget - still - that there is at least one person / being in Middle Earth that the ring has no hold over.
3. The emphasis on battles, and the action sequences. In the Hobbit movies this just went on and on - and so improbable, characters falling for tens of meters and then bouncing up unhurt, like sprites in a computer game. The stories are much more fun and enjoyable - and the adventures more believable because they happen in a context that is in other ways so ordinary, and because they are not over the top.
Sorry realize I haven't really answered the question as posed.
MORE ABOUT THE SONGS
First, most of them are meant to be sung. You write a bit differently for songs than for poems to be read out loud, so I think you miss something by just reading them without music, like, think of them as sung even though they don't have written tunes.
There are a few recordings of Tolkein himself singing humorous songs from the books. So we have tunes we can use for those (though no reason not to make up new tunes for them also).
and
He didn't attempt to sing any of the moving and beautiful songs as far as I know just these humorous ones. It would be interesting to know if he had particular tunes in mind for any of the other ones.
I think also brings out the theme of an earlier world not yet touched by the shadow of Sauron, where people sung in a care free way, just naturally as part of the way they lead their lives, like the singing in Tom Bombadill's house. As Alastair Milne says in comment below (example of "Ho Ho HO"), the darkness then comes over so much more strongly because of that.
Songs all the way through the book like that, either just before the moments of fear and darkness (the Ho Ho Ho), or just after (like the song of Tom Bombadil for old man willow and barrow wight), or in the middle of it.
Even on the edge of Mordor, Sam reciting his poem about the elephant, though not sung, but it's the same kind of thing.
Often you can almost hear the song as you read it. You can try writing down the tunes - I did that, was a fun project, wrote tunes eventually for all the songs in LOR and the Hobbit. The tunes almost wrote themselves, many of them.
I'll add a couple of links to my answer.
This is a list of all the poems in LOR, many of them sung.
Poems in The Lord of the Rings
And this one talks a bit about each one.
About the Songs and Poems in the Lord of the Rings
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
4.8m answer views110.4k this month
Top Writer2017, 2016, and 2015
Published WriterHuffPost, Slate, and 4 moreHuffPost, Slate, Forbes, Newsweek, Business Insider, and Science 2.0