Is there a traditional Chinese New Year greeting?
The most common I have heard is


  恭喜发财 gōng xǐ fā cái


Which is a wish of a prosperous coming year. This is very common in Hong Kong and Taiwan, but I heard that it was frowned upon by the communists on the mainland, since it is a wish that does not align very well with communistic ideals.

One story I read related to this is that there was a legend about a monster called 年 (nian2) that would come down from the mountains around new year and eat people. He is afraid of the colour red, so that's why people would put up red paper on and around their doors. The following morning, when you see someone, you would greet them with 恭喜!; 'Congratulations (to still being alive)!', and then wish that they will 发财; 'get rich', i.e. will have a fruitful and prosperous year ahead.

In Taiwan there is a playful childrens' rhyme related to this. The tradition is that adults give children (without an income of their own) red envelopes with cash, 红包. So the adult says "恭喜发财" and the child immediately (and greedily) replies "红包拿来"; 'Give me the red envelope'.