Tyrion believes that Greyscale can be contracted through touch and through breathing in damp and cold areas. It's also said that Garin's Curse (another name for Greyscale as explained by Tyrion) can be contracted through breathing in fog.
"We'd do well not to breathe the fog either," said Haldon. "Garin's Curse is all about us."
The only way not to breathe the fog is not to breathe. "Garin's Curse is only greyscale," said Tyrion. The curse was oft seen in children, especially in damp, cold climes. The afflicted flesh stiffened, calcified, and cracked, though the dwarf had read that greyscale's progress could be stayed by limes, mustard poultices, and scalding-hot baths (the maesters said) or by prayer, sacrifice, and fasting (the septons insisted). Then the disease passed, leaving its young victims disfigured but alive. Maesters and septons alike agreed that children marked by greyscale could never be touched by the rarer mortal form of the affliction, nor by its terrible swift cousin, the grey plague. "Damp is said to be the culprit," he said. "Foul humors in the air. Not curses."
A Dance with Dragons, Tyrion V
The following passage from The World of Ice and Fire seems to imply the damp fog and waters of Chroyane killed all it's inhabitants of greyscale.
At Chroyane, the cage was hung from the walls, so that the prince might witness the enslavement of the women and children whose fathers and brothers had died in his gallant, hopeless war...but the prince, it is said, called down a curse upon the conquerors, entreating Mother Rhoyne to avenge her children. And so, that very night, the Rhoyne flooded out of season and with greater force than was known in living memory. A thick fog full of evil humors fell, and the Valyrian conquerors began to die of greyscale. (There is, at least, this much truth to the tale: in later centuries, Lomas Longstrider wrote of the drowned ruins of Chroyane, its foul fogs and waters, and the fact that wayward travelers infected with greyscale now haunt the ruins—a hazard for those who travel the river beneath the broken span of the Bridge of Dream.)
The World of Ice and Fire, Ancient History: Ten Thousand Ships