Let $a,$ $b,$ and $c$ be nonzero real numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0.$  Compute
\[\frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}.\]
We have that
\[x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz).\]Setting $x = \frac{1}{a},$ $y = \frac{1}{b},$ and $z = \frac{1}{c},$ we get
\[x^3 + y^3 + z^3 - 3xyz = 0,\]since $x + y + z = 0.$

Then
\[\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} = \frac{3}{abc},\]so
\[\frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2} = \boxed{3}.\]