Now take your derivative with respect to x:

$\displaystyle \Psi ' = A \cdot k_x \cdot \cdot e^{k_x x} \cdot e^{k_y y} \cdot e^{k_z z}$

Now do it again:

$\displaystyle \Psi '' = A \cdot k_x^2 e^{k_x x} \cdot e^{k_y y} \cdot e^{k_z z}$

Now do this for y and z.

Give that a try and let us know how you are doing with it. And yes it looks horrible but most of it cancels out in the end.

If you need to actually solve this, then separate the equation as your hint suggests. You will get equations of the form

$\displaystyle -\frac{\bar /2m}X''(x) = E X(x)$

Solve this for X(x) and do the other coordinate (Y(y) and Z(z)) functions.

-Dan