By Andrew Warinner, Former Editor of the Urban Legend Zeitgeist
Kidney theft is a venerable urban legend. The origins of urban legends are notoriously difficult to identify but in the case of kidney (or other organ) theft, the rumors seem to have preceded the real-life cases by some years.
Here's what is known about the evolution of the stories (and there are multiple versions):
Mid-1980s: Organ theft rumors circulate in Honduras and Guatemala.
In this version, gringos are kidnapping children, harvesting their organs and dumping the bodies (often with macabre notes like "Thanks for the organs!" in English stuffed into the body cavities or bloody cash left in their pockets).
The rumors incited a certain amount of hostility and even violence against Western tourists. Two American tourists were killed by a mob who suspected them of being organ thieves. The U.S. State Department issued a travel warning in 1987 cautioning travelers about the rumors. Some U.S. government officials claimed the rumors were part of a Soviet disinformation campaign.
1990-1: Kidney theft legends hit the U.S. The canonical form does not feature kidnapped, murdered children, this time the organ thieves target hapless single males, something along the following lines: "boy meets attractive girl in bar/girl invites boy back to hotel or home/boy loses consciousness/boy wakes up alone and discovers surgical incision in back/doctors confirm boy is short one kidney." The South America/Central America link is made because the bar is often located south of the border or the attractive girl is often described as being a Latina.
April 1991: Kidney theft is featured in an episode of Law & Order ("Sonata for Solo Organ"). Just from what headlines it was ripped is unclear; it may have been the case of one Ahmet Koc who claimed he was lured to Britain from Turkey by a job offer and woke up after a medical checkup minus a kidney. Or the writers may have picked up on a popular rumor or one its appearances in U.S. newspapers.
1992: Organ theft legends are transplanted to Eastern Europe: orphanages are harvesting the organs of children.
1997: Kidney theft hits the Internet. A widely forwarded email follows the "boy meets girl" form and adds a few juicy details; the victim wakes up in a bathtub of ice, sees "Call 911" written in lipstick on the mirror, calls 911 to find out he is the latest victim of kidney theft. Since the legend circulated by email there is not the usual variation in detail as you see with oral transmission but the location is often changed. Cities featured were Las Vegas, New Orleans and New York; all modern day dens of iniquity.
I saw the 1997 outbreak at first hand; I was writing the "Urban Legend Zeitgeist" at the alt.folklore.urban Urban Legends Archive. It was amazing how widely it spread and how widely it was credited.
Since then there have been several illegal organ trafficking rings that have come to light in Turkey and India.
The Kosovo claims are a bit problematic; it spite of Dik Marty's best efforts, his investigation came up with little concrete evidence, particularly on the transplantation end of the bargains.
It is important to remember that kidney theft rumors were circulating for years before the organ trafficking was exposed; they did not inspire or were the basis of the organ theft rumors and the circumstances of the organ theft do not conform to the urban legends.
So why are organ theft legends popular? The child dismemberment variety taps into a profound fear, maybe one of the deepest we have. The "boy meets girl" variety also ties into to a common fear - one folklorist described them as a "male rape myth."
 The snopes article on kidney theft is good but incomplete -- it doesn't mention the baby parts antecedent in South and Central America and its likely mutation into the U.S. kidney theft tales. Law & Order didn't have much to do with the spread of kidney theft stories; the episode aired after the kidney theft stories were thriving in the U.S.
More questions on pop myths and rumors: