The 'white rabbits' good luck chorus certainly sounds ancient, but the fact remains that it appears nowhere in English print before 1959! Yes, the reference was in the Opies' book about schoolchildrens' lore, but, if it was truly old, one would certainly expect there to have been some reference to it or use of it in earlier literature. The single word 'rabbits' did appear in the same context earlier but not before 1920 either.
There are some who believe it originated with Lewis Carroll's �Alice's Adventures in Wonderland' and others who claim it derives from a German custom of pagan times when pairs of white rabbits were sacrificed at the opening of months for luck.
However, this is one of these things with an air of the historic about it which it does not truly deserve. The "tradition" of saying these particular words on the first of a month for luck is, at most perhaps, about a century old, despite the idea that sacred rabbits existed in much earlier times.