There are a lot of horrible afflictions on Game of Thrones, but greyscale is really creeping up there as being far worse then any sickness — and I'm including being turned into a White Walker in that list. The horrible affliction is a (usually) fatal disease that leaves the sufferers flesh stiff, cracked, and flaking. The skin gets stone-like to the touch, and, unless it's stopped, spreads to the entire body, turning people into "stone men." We all know that Jorah has it, and shared a collective gasp when he shook hands with the Khaleesi, possibly giving the Mother of Dragons greyscale. Jerk move, buddy. We still don't know that much about the disease or many other people who suffer from it, but it's caused fans to wonder if the Targaryens are immune to greyscale.
It's a good theory, and if true would save a lot of lives considering that Jorah may had started a new epidemic by all of the touching. So far the only person we've seen be relatively immune to the 'scale is Tyrion. Many people believe that Tyrion is a Targaryen himself, and thus his not catching greyscale is proof of their immunity. However, even if he were from the house of fire and blood (disclaimer: I do not share this belief; that guy is a Lannister through and through), that doesn't prove anything. We didn't see where the stone man grabbed him, so he could have only touched his clothing. Or, even if he touched skin, Tyrion could have not contracted it by chance.
Though the show hasn't touched on any of the Targaryens catching it, we also haven't seen anyone from House Stark, or House Lannister, or, heck, even House Tully get it. Are they all immune? I doubt it.
This was never directly addressed in the books, except for a very offhand comment in A Dance With Dragons where Daenerys mentions her brother Viserys saying Targaryens were "untroubled by the pestilences that afflicted common men." This is a really nice sentiment, but I've got a few major problems with it. First off, I don't think the body-calcifying effects of greyscale can be considered common; this isn't exactly a flu you recover from after a month. Second, this clearly did not extend to mental illness since the Targaryens are well known for that. Finally — consider the source, people. This is Viserys we are talking about, who was unstable at best and often inflated his own importance and the value of their blood.
It would be nice if they were immune. I'm rooting for Daenerys to sit on the Iron Throne, and (despite a rocky last season) I want her to have all of the advantages she can get. Being immune to colds, fire, and greyscale would certainly give her a leg up on the competition as everyone goes to war. I also don't want her inadvertently starting a plague all because Jorah couldn't keep his hands to himself. However, for now I think this rumor is just that, an unfounded rumor. Prove me wrong George R.R. Martin!