Generating multivariate normal samples - why Cholesky?

After the comment of Rahul you understood that in any parametrization $x=Av+μ$ you will need that $$ Σ=\Bbb E(x-μ)(x-μ)^T=A·\Bbb E(vv^T)·A^T=AA^T. $$ There are infinitely many possibilities to chose $A$, with any orthogonal matrix $Q$ also $\tilde A=AQ$ satisfies that condition.

One could even chose the square root of $Σ$ (which exists and is unique among the s.p.d. matrices).

The advantage of using the Cholesky factorization is that you have a cheap and easy algorithm to compute it.