There's an Irish traditional called 'Avon Dale', which I associate with the beauty of Middle Earth scenery, and this part of this song here:
'(...) pride and ancient glory fey,
such was the land where he was laid"
brings to mind the battles and lastly, Aragorn's death in Minas Tirth.
The song is so nostalgic and beautiful, it sounds like something straight out of Middle Earth.
And there's this Christmas song which has absolutely nothing to do with Middle Earth, that makes me think "Minas Tirth!" only because there's a line there:
"White city, she's so beautiful!' 
The song is 'December will be magic again' by Kate Bush.