Romanian singer Anda Adam said she was turned back from a border point late Saturday with her band members.
Other Romanians were reportedly told they could not prove they do not have AIDS.
Some two thirds of Moldova's 4.1 million people are ethnic Romanian and there are tensions between the countries over fears that Romania is seeking more influence.
Since Romania joined the European Union in 2007, thousands of Moldovans have received Romanian passports.
Romania's Foreign Ministry said Friday it sent a letter of protest to its Moldovan counterpart asking for an explanation.