Python 3, 126 116 112 bytes

Edit: I'm keeping the other answer because it's cool, but this is shorter.

from time import*
"Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Goat".split()[localtime().tm_year%12]

If trailing whitespace is ok, then it's 126 bytes. If not, it's 134 bytes after adding .strip() to the end of the code.

from datetime import*
"MRDPROTRDSHGoooiaxiarnoonoggt gbaaraks    ebgkstet    rioee ye     tn    r"[datetime.now().year%12::12]