The wine region is in the southeastern corner of the country, about an hour’s drive from the second-largest city in Slovakia, Kosice. The easiest way to get to the villages from Kosice is by train or rental car. I chose the latter option, although the train might be a bit easier for a first-time visitor. Compared with the much larger and much more developed wine region in Hungary, the Slovakian wine country keeps it simple — and affordable. “We want to offer a more personal approach to visitors,” Mr. Macik said.

“We don’t want to commercialize it too much; there are enough wineries like that elsewhere,” said Ms. Kapralova. And since it’s not very practical to explore the wine country as a driver, I decided to continue my three-day trip on a bike, rented at the Tokaj Macik Winery. From Cerhov, which has a small wine history museum, I took the road up toward the vineyards and wine cellars, savoring the sunny weather.

It proved to be a rewarding travel option — although sometimes challenging on the hills — with my favorite stop being the new viewing tower in the shape of a wine barrel in the vineyards above Mala Trna.

Accommodations in the area tend to be low-key. The wineries J. & J. Ostrozovic in Velka Trna, Tokaj Macik in Mala Trna and Zlata Putna in Vinicky all have their own small hotels. Historic cottages, however, can be an interesting option for the travelers looking for cozy authenticity. One of them, called the Cottage, appropriately enough, is in Velka Trna, overlooking the village and green vineyards from a tranquil hill. The 100-year-old white house with a welcoming porch is decorated with antiques, vintage hand-painted plates and a kitchen counter made of old red bricks. Ms. Kapralova, the winemaker, rents out the cottage, which can accommodate up to 13 guests, as part of a small family business. She lives next door with her family, and is on hand to help with anything — from wine cellar visits to cooking dinner.

Apropos of dinner. Although there are not many restaurants in the villages, several of the wineries serve delicious regional cuisine. The owners will happily bring you holubky, a meat-and-rice mix wrapped in a cabbage leaf (my personal favorite), the fresh pork specialties or lokse, salty pancakes paired with meat as well as sweet fillings. The Zlata Putna restaurant serves homemade chicken soup, goose liver specialties and fish from the nearby Bodrog River.