The Georgian province of Kakheti, between the mountains and the Black Sea, has charming small towns, idyllic countryside and an ancient winemaking tradition that’s just starting to catch a buzz.

IT WAS 9:30 in the morning and I was already tipsy. At Twins Old Cellar Winery Hotel in Napareuli, a one-road village of stone houses and overgrown pomegranate trees in the Georgian province of Kakheti, breakfast came with an all-but-mandatory free wine tasting. For Eldar Ramishvili, the hotel’s ebullient manager, a taste meant three full glasses.

Traveling through Georgia, the tiny post-Soviet country set between the Caucasus and the Black Sea, is always a metabolic endurance test. Wine, brandy, chacha—a grape-skin moonshine with the flavor of gasoline schnapps—all these are habitually, exuberantly, foisted upon any foreigner who sits still long enough. But in the country’s primary wine region of Kakheti—according to Georgians, the birthplace of wine itself—consumption seems to be the primary occupation.

On the bright side, I’d been assured, I didn’t have to worry about hangovers. Georgians hold that wine fermented the traditional Georgian way, in a qvevri (a 1,000-liter beeswax-coated terra-cotta jar buried in the earth), triggers no aftereffects. “It’s the old Georgian way,” Mr. Ramishvili explained, showing off the hundred qvevris that lined the winery floor. Fermenting wine “the European way” (in tanks, aging it in oak barrels) makes it “so…light. For appetizers.” He gave a dismissive sniff. Qvevri white wine is sharp, strong, amber in hue—its color absorbed from the clay—or in the case of reds, so dark it’s known as shavi gvino: black wine.

For thousands of years (the first qvevri shards date back to the 4th millennium B.C.), Kakheti’s winemaking was largely a family endeavor. Even now, nearly every Kakhetian I met boasted of his personal backyard qvevri. Relatives who live in Tbilisi, Georgia’s capital, some 60 miles away, still come out for the weekend with empty 5-gallon plastic jugs, which they’ll fill up from auntie’s backyard reserves before heading back home.

But wine is becoming less of a family affair in Kakheti and more of a commercial enterprise. Tourism has been on the rise throughout the country over the past half-decade, turning this picturesque province into a would-be Burgundy of the Black Sea. While the climates of Georgia’s lowlands and the south of France are similar in terms of temperature and rainfall, Kakheti’s rust-colored soil brings a dark earthiness to the region’s wines.

“Many Georgian wines may lack the sophistication of wines produced with the help of modern technology,” said Carla Capalbo, a food and wine writer, but at their best, “qvevri wines are like a pure expression of the fruit uncluttered by layers of oak, toast or tricks.”