I really don’t have any attachment or love of traditional Christmas music, especially performances that go back a long way. My wife and I have tried a few newer ones and are wondering what people like. I will make one recommendation.
Tommee Profitt - Birth of a King
It’s a great album and even better, he just uploaded a full live concert of it to Youtube. A very nice thing to put on in the house.
The very few Xmas songs (not album) I have in my collection are tolerable when they come up on iPod shuffle, but I don’t seek them out.
Here’s an album I wouldn’t mind having.
My wife has a couple of old Christmas albums by the big band Fred Waring and the Pennsylvanians, and those get taken out every year.
I have a Christmas CD by Kristin Chenoweth, from 2008, which I enjoy listening to every year, as well.
Years ago I bought my SIL a Putumayo CD of world music Christmas songs. She plays it all the time over the holidays. I went to look it up and, although I didn’t spot the album I bought decades ago, it seems that you can just stream all their stuff in multiple formats:
Mistletoe Jam
the Christmas Jug Band (Dan Hicks and others)
including my favorite Xmas Carol:
“Somebody Stole My Santa Claus Suit”
An American Christmas. It was put together by a professor of American music who lamented that so much of our country’s musical tradition has been forgotten, and decided that a Christmas album was a good way to raise awareness. There are a few old classics on there that anyone will recognize, but a lot of songs (good ones, too), that I’d never heard anywhere else.
@Chronos I’m intrigued, so I tried to Google this. Do you have a little more info about this? There are a lot of albums (and sheet music) with that name, and adding stuff like “professor of American music” isnt helping.
I don’t know if you’d consider it “non- famous,” but it’s relatively recent and has no traditional Christmas songs on it: Sia’s "Everyday is Christmas. ’
I’ve grown quite fond of it over the past few years. I have it on in my truck quite often this time of year.
Here are all 33 songs:
Barenaked for the Holidays. Most of the songs are covers of traditional music, but this counts, right?
Why thank you.
It turns out I already have “Christmas In Jail”. 
There are bagpipes.
Squirrel Nut Zippers’ “Christmas Caravan”. I absolutely love it. Made a Pandora station based on this album and the music that comes on is equally hot!
This album is great! So many cool original songs and unique twists on old ones.
Came here to mention the Squirrel Nut Zippers — glad to see I’m not the only fan
“God Rest Ye Merry Gentlemen”/“We Three Kings” is quite popular on Canadian radio stations this time of year.
I remember hearing on the radio a guy singing a cover of Joni Mitchell’s “River” that I thought was pretty good. Turns out it was Robert Downey Jr! (From the album “Ally McBeal: A Very Ally Christmas”)
Sarah MacLachlan’s Christmas album “Wintersong” is very very good, in my opinion.
Jazz
Louis Armstrong and Friends, ‘The Best of Christmas Songs
Blues
Ella Fitzgerald, ‘Ella Wishes You A Swinging Christmas’ (1960)