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I read somewhere that the year of the rat is associated with pork, peas and cabbage. So I put together a list of recipes with these ingredients.

Hillary
Chew on That

I don't know of any foods that are particular to any year of the zodiac, but here are some traditional foods enjoyed during the New Year
...this is a late response, but the New Year celebrations don't just end on New Year day!

- nian gao...literally "year cake" but if you like wordplay, I guess it could also sound like "sticky cake" which it is! When fried in a pan in a little oil (sometimes with a light egg batter if desired), this gummy brown sugar and rice cake mixture turns molten-sticky and just barely contained by its light crust. You can also steam it, but its not nearly as yummy or fun!
This is usually only available and eaten around the New Years. (I have to get my fill now...)

- fa gao...light, steamed rice flour and brown sugar cupcakes. Its rapid and fluffy rising and expansion in the steamer or "fa" symbolizes rapid growth of wealth and fortunes.

- zai...this might only be a tradition from the village the Cantonese side of my family came from, but the first meal of the New Year is always vegan, as a sign of respect towards our agricultural roots, and also to start out the New Year on a clean and peaceful note.
A mixed dish of mushrooms, mung bean sprouts, rice noodles, yuba tofu skin and pulled tofu in a savory broth spiced with fermented tofu (fu yu) usually takes center stage of the meal, along with steamed greens, a tofu dish or two, and always a sweet soup with more tofu skin in it to finish off the New Year breakfast or brunch.

- round foods (usually symbolized by deep fried pastries like sesame balls, or tang yuan which are just small balls of chewy mochi in sweet soup) to symbolize the round shape of money, also unity and familial togetherness.