Almost every corner of the world produces its own take on the vampire myth, but South Eastern Europe leads the pack. From Romania down to Greece there's myriad local variants on the blood-sucking undead and evidence that these myths persist well into the 20th Century (or in Romania's case, well into the 21st!).
But what was it about the region's history - it's culture, religion, folklore and conflict - that gave birth to so many vampire tales, and why was there an 'epidemic' of reports in the 17th and 18th Century?
This isn't a question about the things that people saw as evidence of vampirism (bloating corpses, etc) specifically, but I appreciate it may play a part if burial customs are a factor.
I would actually argue, that the Balkans did not have proportionally more "vampire myths" than other (European) regions - they are simply better documented.
First of all, we need to figure out how the vampire myth fits into the regular old undead pantheon. And, right off the bat, the big letdown: The modern vampire we have in present in our minds thanks to the various medias has very little to nothing to do with the "vampire" that supposedly scoured and still scours the Balkans. The classic Balkan vampire is much more akin to the German "Wiedergänger" (lit.: "Walk again(er)") - something that would closely resemble a Zombie in modern media (which, in turn, has little to nothing to do with the Vodoo Zombie from which it lends its name).
But why do we have such a strong connection of the Balkans with vampires and the vampire myth?
Well, it surely has been present in South-Eastern Europe for a long time, as evident by (for example) the report of Father Leo Allatius, who reported to the Vatican in 1645 of the happenings on the Greek island of Chios:
"For very often, inhabiting this body, he [the devil] comes forth from the grave, and going abroad through villages and other places where men dwell, more especially at night, he makes his way to what so ever house he will, and knocking upon the door he calls aloud by name in a hoarse voice one who dwells within. If such a one answers he is lost; for assuredly he will die the next day. But if he does not answer he is safe. Wherefore in this island of Chios all the inhabitants, if during the night they are called by anyone, never make reply the first time. For, if a man be called the second time it is not the vrykolakis who is summoning him but somebody else."
[Quoted after: Summers, Montague: "The Vampire in Europe" p. 224]
But for the Balkans, another factor was much more important. In 1718, the Austrians triumphed over the Ottoman Empire and the Treaty of Passarowitz was signed. With it, Austria-Hungary gained control over much of modern Serbia, some parts of modern Bosnia as well as Wallachia - part of modern Romania. And the Austrians, in light of the dawning modern era, were keen on thoroughly researching their new lands. And when reports of undead returning from their graves to haunt the living began to trickle in (sometimes paired with requests to be allowed to dig up a corpse and stake it so it may stay dead), the Austrians sent clerks to document the cases. And this gave us the first really documented cases of "vampires".
The most prominent case is that of Peter Plogojowitz. In 1725, a case of a "wiedergänger", or strigoi as they were called in the Balkan region, was reported to the authorities. In the report, we can find accusations that the dead Plogojowitz came to his former house and murdered his son when he refused to give him food. The peasants then dug him up, found fresh blood on hips lips (most likely a sign of decomposing) and subsequently saw their fears proven right. They staked the corpse, burned it and reburied it.
[An account of this story is also found in Summers, but the original Austrian report can also be accessed here in German]
This report spread throughout the Holy Roman Empire and from there through Europe and sparked a regular "vampire hype".
There were some similar cases later on (for example that of Arnold Paole), that by far overshadowed remarkably similar reports from Bohemia and other parts of (predominantly Eastern) Europe and cemented the picture of the "mysterious Balkans" being haunted by the undead vampires.
The Balkans at this time were the "twilight zone" between Orient and Occident and were seen as rather backwards. The vampire myth perfectly fit this picture and this view on the Balkans might very well (at least in my opinion, as the only book concerning this topic came out after I did my research on the topic and I've yet to read it) have significantly contributed to Bram Stoker creating Vlad Dracula as coming from the Balkans. The weird, backwards area between the West and the ever-threatening East. But Stoker certainly further cemented the cultural connection of the Balkans and vampires.
Sources (among others, which are mostly German, so I will for now only post my English language sources):
Barber, Paul: „Vampires, Burial, and Death. Folklore and Reality.”, Yale University Press, New Haven and London 1988
McNally, Raymond T. and Florescu, Radu: „In search of Dracula: The history of Dracula and vampires.”, Mariner Books, Boston 1994
Summers, Montague: "The Vampire in Europe", Kessinger Publishing, Whitefish 2003
evidence that these myths persist well into the 20th Century (or in Romania's case, well into the 21st!).
Sorry, what do you mean by this?
The myth itself existed all over Eastern Europe, but it got "popularly" assigned to Transylvania. The vampire myth originates from a human condition called porphyria which in short gives the individual high amount of sun sensitivity (therefore no tanning, being indoors unless night) and high iron deficiency (therefore the need for food high in iron, like animal blood). The porphyria itself became a common condition among the Eastern European nobility because of the rampaging incest (it is a genetic disease in other words). And it got to be popularized to Western Europe as Transylvanian in origin by German merchants from Transylvania that got persecuted by the Wallachian prince Vlad "the Impaler"