jmc

$\text{Solution 1: Find the first term and the common difference.}$

Let the first term of the sequence be $a$ and the common difference be $d.$ Then, the third term is $a+2d$ and the sixth term is $a+5d,$ so we have the system $a+2d=5,$ $a+5d=-1.$ Subtracting the first equation from the second gives $3d=-6,$ so $d=-2.$ Substituting this into either of our original equations gives $a=9,$ so the twelfth term of the sequence is $a+11d=9+11(-2)=\boxed{-13}.$

$\text{Solution 2: Use our understanding of arithmetic sequences.}$

The sixth term is $6$ less than the third term. The twelfth term is twice as far from the sixth term $(6$ steps$)$ as the sixth term is from the third term $(3$ steps$).$ Therefore, the twelfth term is $2\cdot 6=12$ less than the sixth term, so it is $-1-12=\boxed{-13}.$