The most common Chinese ways of saying Happy New Year are Gong Xi Fa Cai (Mandarin) and Gong Hey Fat Choy (Cantonese). Even though the pronunciations are a little different, both are written the same way.
Traditional Chinese: 恭禧發財; Simplified: 恭禧发财.
Gong Xi (恭禧) is congratulations or respectfully wishing one joy.
Fa Cai (發財) is to become rich or to make money.
Thus, Gong Xi Fa Cai means wishing you to be prosperous in the coming year.
A fun way to respond to someone who greets you with Gong Xi Fa Cai is Hong Bao Na Lai, "Red envelope please!"
Chinese also have a literal way to say Happy New Year in Chinese.