Raining Cats and Dogs

Well, the Historian/Writer Eric Sloane wrote in his book 'Folklore of American Weather' (1963, Hawthorn Books, NY), "This is believed to be a German mispronunciation of "cats and ducks." The Pennsylvanian German people used to say, "It is raining to keep in the cats and bring out the ducks." They also said, "It is snowing for cats and ducks," which meant that the snowfall was sufficient to track a cat or a duck through it."
This makes a bit more sense that animals falling off roofs, I should say.

More believable than animals falling off roofs certainly, but that isn't saying much, as that story is clearly nonsense.

'Cats and ducks' adds to the numerous theories. Presumably though, if Sloane had any evidence for it he wouldn't have resorted to 'it is believed'.