# Python 3, <s>126</s> <s>116</s> 112 bytes

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**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]