jmc

We can rewrite $z^2=12z-7$ as $z^2-12z+7=0$. Because the sum of the roots of a quadratic is $\frac{-b}{a}$, we know that the sum of all values of $z$ such that $z^2-12z+7=0$ is $\frac{-(-12)}{1}=\boxed{12}$.