To solve this problem, I will use some algebraic identities and some clever manipulations. Since $a + b + c = 0,$ the first term is zero, and I get $a^3 + b^3 + c^3 = 3abc.$ This means that the fraction simplifies to $\frac{a^3 + b^3 + c^3}{abc} = \frac{3abc}{abc} = 3.$ So, the only possible value of the fraction is 3, regardless of the values of $a,$ $b,$ and $c.$ 
Let $a,$ $b,$ $c$ be non-zero real numbers such that $a + b + c = 0.$  Find all possible values of
\[\frac{a^3 + b^3 + c^3}{abc}.\]Enter all the possible values, separated by commas.