Set in the 1950s, Vienna Love follows Ryan Barger. He is a self-described “hillbilly hick” from a log house in Mulybuck Holler, Kentucky, who is stationed in Salzburg after being drafted into the U.S.

Some fall quickly for the charms of cities like Paris and Rome. Such destinations wear their hearts on their sleeves, with well-known attractions drawing in the crowds. Other destinations don’t flaunt ...