Although some may use flashcards on DS systems as a convenient way to store copies of games they own, many people use them to play games they have not bought. Piracy was a big problem for the DS family of systems, and Nintendo always reiterated that it would ensure that 3DS was more secure, closing the door on illegal copies of games.
So far these efforts have been a success, with hacks of 3DS software and homebrew apps still not in existence despite the handheld being on the market for 18 months. That said, many DS flashcards have worked on 3DS, but it's emerged that last week's system update, 4.4.0-10U, has stopped many of these from working. According to forum posts dug up by tinycartridge.com, an administrator on SuperCard's forum has come out and said that, by their estimation, 95% of DS flashcards are blocked on 3DS with no software workaround. For the cheeky scamps affected this means waiting for these companies to revise their hardware and re-release it.
Some DS flashcards do still work on the latest handheld, apparently, but Nintendo is no doubt looking at future updates to completely close the door. After a long period of keeping 3DS software locked down, it seems that the big N is now ensuring that the legacy of DS copies will end in this generation.