Is Classic Country off the table? I see you said yes to Johnny Cash.
Hank Williams
Loretta Lynn - Fist City is a lesser known gem of a song she wrote herself
Glen Campbell - nothing whiter than Wichita Lineman or By the Time I Get to Phoenix
Conway Twitty - I Love to Lay You Down, anyone?
And if we are going Old World, Loreena McKennitt’s early work is very Celtic. Her later stuff is more Middle Eastern influenced.
If none of this is White enough, then shit…Ray Conniff Singers?
Oh, and Mac Davis’ Hard to Be Humble should go a long way to lifting you up when you are down. 
I think it’s White cool, anyway…
Last add-on, I promise!
Jerry Reed - Amos Moses, She Got the Goldmine I Got the Shaft, East Bound and Down
He’s so Cornbread Cool in my book!
Weird Al when he does that one straight up polka song on each album.
Lots of They Might Be Giants should count.
Also I’m going to go out on a limb and say anything by Tiny Tim.
Aye; it was a joke. But it also wasn’t.
AWB are from Scotland. In the early 1970s. I think that could fairly be described as “white”. And the kind of music they make is far removed from their native culture and local influences, yet they totally fucking rock. They are better than most bands made up of people who grew up where funk and R&B were the native culture.
So no, their music isn’t free of jazz or blues influence (let’s face it; that’s like +/- 94% of all accessible music in the 20th and 21st centuries), but they were white people who made cool music that showed that culture has far more influence than race.
People who were cool 30-35 years ago do.
Seriously, we’re a bunch of mostly older types here, who among us knows what’s cool these days? ‘Cool’ in music is pretty much defined by teens, twentysomethings, and those who cater to their tastes.
But the first thing I’d do to narrow it down is change your all-caps text to NOTHING PRE 21ST CENTURY.
ZipperJJ mentioned Radiohead earlier on. That may be the only band mentioned in the thread so far that’s most known for songs released in the new millennium. I don’t know if Radiohead is still cool (I’m 63, how the fuck would I know?), but they might still be cool. It’s hard for me to imagine that any other band or artist mentioned in this thread is still cool.
Hmmm…Cage the Elephant, the Lumineers, Florence + The Machine. All recent, all pretty white both in skin pigmentation and musical style.
Need some young folks to tell us whether they’re cool or not, though. (I like them all, but that is unquestionably NOT evidence in support of their coolness. Possibly the reverse.)
Bjork?
The Velvet Underground?
The Smiths?
The Stone Roses?
Brian Eno?
Early Roxy Music?
Problem is, I have the nagging suspicion that if I think something’s cool, it probably isn’t.
It’s true; I am a bad, bad man. Still pleased that, with over 60 replies, you’re the only one who’s said that, and I have no idea who you are.
Everyone citing polka music, you ARE being “ironic,” aren’t you? Or did polka slide right over my head into the world of cool things?
Young Jeb! on lead guitar.
You don’t hear any african american in Jerry Reed? Amos Moses was like a minstrel show song. When you’re hot you’re hot? Same thing. He was working that angle like a MF.
On the whole saying that country musicians aren’t black influenced is ahistorical. It was a thread that ran through the whole forming of the musical style, in one way or another.
I would say a good portion of the artists mentioned would still qualify as “cool.” I mean, okay, I’m 42, but tastes are definitely more diverse now, and the 20- and 30-something I know who are into music still find much of the music in this thread cool.
Lou Reed is kind of known for singing blackly. All those street songs didn’t come from white cultural traditions.
White and cool and 20th/21st century? Stravinsky, Schoenberg, Puccini, Prokofiev, Shostakovich, Bartok, and many more such. White music rocks!
And outside the classical tradition, as clarified later in the thread.
Karlheinz Stockhausen is my ‘for serious’ nom.
WordMan mentioned Steve Reich, another favorite of mine.
Many can’t stand Phillip Glass’s work, but I think the guy creates awesome music.
Also, Rodrigo y Gabriela have had a lot of success the past few years.
Ronnie Drew and Luke Kelly and the Dubliners were cool.
Classic 70s rock.
White people make good music. It’s all the mayonnaise and microbrewed beer we eat, it keeps our brains running on full throttle.
I would say most 70s rock is just dripping with blue influences.