My wife was given a ten euro bill as part of the change from a purchase at a retail shop in Athens. Moments later when she tried to pay for another item (at a open air antique stall) the antiques vendor held the ten euro bill up to the sunlight and told us he would not accept it bbecause it was counterfeit. We then asked a nearby uniformed policeman/security guard if the bill was invalid. He, too, held it up to the bright sun and agreed that it was not a valid bill. When we confronted the shop vendor who had given us the bogus bill, she protested but eventually exchanged it for a ten euro bill which another vendor accepted.
My wife recalls that she thought it was odd that the offending vendor asked my wife during her initial visit to the shop to go downstairs-presumably away from direct sunlight-to receive the initial ten euro counterfeit bill. From this episode, we concluded that the bogus bill lacked a watermark (or equivalent) which can best be seen when brightly back lit.
While this was the only even slightly unsavory moment in a marvelous ten day trip to glorious Greece, we doubt this was an isolated incident because the local antiques vendor knew to check for the validity of the bill as if he had been through this process more than once. We recommend you get coins, not bills, from vendors if you transact in currency.