According to anime, from whence I get all my wordly knowledge, it is considered a Good Omen if you dream about certain things as your first nocturnal imaginings of the New Year. The list, obtained from my handiest source (Lucky Star, natch), goes, in order: Mount Fuji, a hawk, an eggplant. Miwiki Miyuki adds a fan (as in a cooling device, rather than an obsessive), tobacco, and a blind man. You do not have to dream these in sequence or frequency, although the more items you dream of, the luckier you will be.
That said, I am uncertain what the year will bring when my first dream of 2009 involves a samurai dwarf king with a jazz band.