Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.
If you define $p=X^n-1\in\mathbb C[X]$, then $p(A)=0$. This tells you that the minimal polinomial $m_A$ of $A$ divides $p$ and, in particular, that $m_A$ has all its roots simple, because the same is true of $p$.
It follows that $A$ is diagonalizable, so, up to similarity, you can suppose that it is diagonal.
Can you see which are the diagonal matrices $A$ which satisfy the condition $A^n=I$?
NB: This argument does not depend on your knowing about Jordan canonical forms.