It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
The following equivalences are known, and not very useful:
$L^1(\Omega,A, P)$ is separable
for all $p \in [1,\infty)$, $L^p(\Omega,A,P)$ is separable
the $\sigma$-algebra $A$ of measurable sets with the pseudometric $\rho(a,b)=P(a\Delta b)$ is separable, where $\Delta$ is the symmetric difference.