I live in Poland, but having lived 17 years in Texas I love country music and try to stick some country songs into my pop bands' sets. Of course you can't do "Screw You We're From Texas" or probably even Roger Creager "Everclear" because few people will get the cultural references, but some stuff does work very well.
Some that I've found that work:
"Need You Now" by Lady Antebellum (popular enough to get radio airplay here)
"Homegrown Tomatoes" by Guy Clark
"Wagon Wheel" by Old Crow Medicine Show (I'm really surprised this one gets good responses, but due to lack of banjos and fiddles we do a pretty modern version and I do my best Colt Ford imitation on the third verse)
Work OK but better with more foreigners in the crowd:
"Crazy Ex-Girlfriend" by Miranda Lambert
"Hemingway's Whisky" by Guy Clark
Throw some more ideas at me. I'm thinking funny or romantic songs would be easiest, basic stuff that everyone gets immediately even if they've never heard the song before or their grasp of English isn't perfect. I'm probably not gonna do a full country project anytime soon, but it would be great to have a solid set's worth.
If anyone's curious how we do them we've got most of these at
my Youtube channel, except for "Need You Now" because the camera always seems to run out of battery before we get to it, haha. Too bad though, it's such a great song.