The integral of sin(2*Pi*x) from x=a to x=b is zero if and only if b-a is an integer. Similarly, using Fubini's theorem, the integral of f(x,y,z)=sin(2*Pi*x)*sin(2*Pi*y)*sin(2*Pi*z) over a box is zero if and only if one of the box's side lengths is an integer.
So suppose you are given a box which has been subdivided into smaller boxes, and each smaller box has a side of integer length. Integrate f(x,y,z) over the large box. Then this is the same as the sum of the integrals over the smaller boxes. But all of these are zero, meaning the integral over the large box is zero. QED
Remark 1: This proof works with very little modification in n dimensions.
Remark 2: Jurek Dydak has found a combinatorial solution to this problem for rectangles in the plane.