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