jmc

Because the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four congruent right triangles. The legs of one of these triangles measure $12/2=6$ and $16/2=8$ units. Therefore, the hypotenuse of each triangle measures $\sqrt{6^2+8^2}=10$ units. Since the side length of the rhombus is equal to the length of the hypotenuse of one of the triangles, the perimeter of the rhombus is $4\cdot 10=\boxed{40}$ units.