First of all, both L and p are vectors, so the commutator should be computed componentwise. Next, you need to find a common dense everywhere domain for the commutator, it's not difficult to see that on the Schwartz space over R^3 both the momentum and the angular momentum operators are essentially self-adjoint and the invariance conditions are met. Therefore,
[tex] [L_{i},p_{j}]_{-}\psi (\vec{r})=... [/tex]
and , without doing any specific calculations (derivatives i mean), using the fundamental comm. relations (also valid on the Schwartz space) and some simple Levi-Civita pseudotensor manipulations, you can find the answer.
I assume you know how to take a cross product. The only other thing is that p = -i\hbarh\del which acts on the wavefunction \Psi, and you can't exchange r and p (ie. rxp is not the same as pxr)