jmc

We have $f(3m)=(3m{)}^2+12=9m^2+12$, so $f(3m)=3f(m)$ gives us $9m^2+12=3(m^2+12)$. Expanding the right side gives $9m^2+12=3m^2+36$. Simplifying gives $6m^2=24$, so $m^2=4$. Since we are given that $m>0$, we have $m=\boxed{2}$.