Adding most of these to my Spotify list now. Keep 'em coming!
Eddy Arnold --145 country chart appearances in 38 years. Take your pick.
Hank Thompson – What’s Made Milwaukee Famous and the very similar New Records on the Jukebox, both from his LP Smoky the Bar.
Tex Williams – Smoke! Smoke! Smoke! That Cigarette
Red Foley – Tennessee Border
Also, you might want to check out a contemporary artist who channels Hank Williams uncannily: Wayne Hancock.
The three lesser known songs Elvis sings in “Love Me Tender” all have a wonderfully strong country flavor:
Country is not my favorite
but when I was growing up my best friends’ mother was from Tennessee
Make Believe by Conway Twitty
Anything by Johnny Cash Ring of Fire was a favorite
Ode to Billy Joe by Bobbie Gentry
Deeper Well Emmylou Harris
“There Stands the Glass” by Webb Pierce
“I Can’t Stop Loving You” and “Oh Lonesome Me” by Don Gibson
Saw him do this in concert once. Awesome. It might have helped that I was close to the speakers – my whole self was sorta vibrating.
Love this one too. Used to play it at grandma’s house.
I think George Strait can be considered classic country. The Cowboy Rides Away is my favorite of his. I like that it was on the jukebox in the bar scene in Near Dark. 
For The Good Times, Ray Price
He’ll Have to Go, Jim Reeves (worth mentioning again)
Mississippi Cotton Pickin’ Delta Town, Charley Pride
My personal favorite: Good Time Charlie’s Got The Blues, Danny O’Keefe
Amos Moses, Jerry Reed
Mr. Sandman, Chet Atkins
Eddy Arnold had too many to list: the man had a hell of a voice. One of my faves is The Cattle Call
Absolutely, George Strait is classic country. There are a few performers out there today who are doing true classic country (unlike most of the Nashville junk that the great Robbie Fulks called “soft-rock feminist crap” in his takedown of Nashville, “F**k This Town”).
Randy Travis, before his current troubles, recorded some great stuff. “On the Other Hand,” “Digging Up Bones,” and more. And his gospel stuff is great.
The above-mentioned Robbie Fulks has a zillion great ones. “The Buck Stops Here” is truly a country classic.
Dwight Yoakum, especially early in his career, has put out some great stuff.
Yes. And where would the Everly Brothers have been without the Louvins?
Here are a couple of funny country songs that fit your criteria: “They Ain’t Making Jews Like Jesus Anymore” by Kinky Friedman and “Last Meal” by Asleep at the Wheel.
Well shoot, if we’re tossing the 1980 criteria if it’s a country-esque comedy track (since Last Meal was released in 2001), let’s see what the OP thinks of The Austin Lounge Lizards?
A few of their hits: Teenage Immigrant Welfare Mothers On Drugs. Leonard Cohen’s Day Job. The Dogs, They Really Miss You. There are a bunch of others, and they are great live.
Hard to go wrong with Kinky, though.
I am loving the recommendations in the thread.
Tom T. Hall, Fox on the Run .
Mickey Gilley - Headache Tomorrow (or a Heartache Tonight).
My mom told me once that Mickey Gilley wanted to date her, but she wouldn’t have anything to do with a musician. She said she thought “He probably wouldn’t make anything of himself.” He wound up marrying a coworker of hers that he asked out after my mom turned him down.
:smack:How could I forget The Girls All Get Prettier At Closing Time?
There are some great suggestions here. I’ll add a few that I like:
The Jet Set George Jones and Tammy Wynette
Louisiana Woman, Mississippi Man Conway Twitty and Loretta Lynn
Just Someone I Used to Know Porter Waggoner and Dolly Parton
Always Patsy Cline
Four Walls Jim Reeves
Waltz Across Texas Ernest Tubb
Mule Skinner Blues Dolly Parton is my favorite cover
Paper Mansions Dottie West
Country Bumpkin Cal Smith
Anything by Lefty Frizell, Willie Nelson, Johnny Cash et al
Look up the Real Country radio network. My local affiliate plays about a 50/50 mix of classics and top 40 country in their country rotation. You come across some really good ones.
There are some current artists who belong on this list as well. Robbie Fulks was mentioned, but there is also BR549, Jamey Johnson, Wayne “The Train” Hancock, Chris Sprague, The Sprague Brothers, Drag the River, John Doe (of X) and the Sadies and SCOTS just to name a few.
Double thumbs up for Robbie Fulks! “She Took a Lot of Pills and Died” is one of my favs, along with “North Carolina is a Cigarette State” and “Every Kind of Music But Country.” “Barely Human” if you want serious depression-good country. 
Modern Sounds in Country and Western Music by Ray Charles. Buy it now. I’ll wait.
Don Gibson has been mentioned; don’t forget “Just One Time” and “A Legend In My Time.”
“Please Help Me I’m Falling” and “Send Me the Pillow You Dream On,” Hank Locklin.
My dad was a big Country fan, so I grew up with his vast collection of 8-tracks (!). There was plenty of Merle Haggard, George Jones, etc., but the particular artists that have stayed with me through my Rock ‘n’ Roll rebellion are Hank Williams (Sr. of course), Johnny Cash, and Charley Pride (who I always thought was one of Country’s lesser-known lights; glad to see him mentioned here).
Some favorites:
Hank Williams:[ul][]Move it on Over[]Mind Your Own Business[]Long Gone Lonesome Blues[]Why Don’t You Love Me[]Half As Much[/ul]Johnny Cash:[ul][]I Walk the Line[]Ring of Fire[]One Piece at a Time[]There You Go* Riders in the Sky[/ul]Charley Pride:[ul][]Before I Met You[]Just Between You and Me[]I Know One[]Too Hard to Say I’m Sorry[]All I Have to Offer You is Me[/ul]