Euler found a brick with integral diagonals on the faces, when the lengths of each side are 44, 117, 240. I too have found this solution.
Euler didn't manage to find a brick with integral cross diagonals (i.e. w*w+h*h+d*d).
And guess what? Neither did I! All the way up to 1000x1000x1000.
But I did manage to find some other examples of Eulers problem. Alas, none were smaller than the original solution.
44 x 117 x 240 (Euler's original)
85 x 132 x 720
88 x 234 x 480
132 x 351 x 720
140 x 480 x 693
160 x 231 x 792
176 x 468 x 960
231 x 160 x 792
240 x 252 x 275
252 x 240 x 275
351 x 132 x 720
468 x 176 x 960
480 x 504 x 550
720 x 756 x 825
Not that anyone needs telling, but since I'm unlikely to be around to blog it on Sunday I'll point out this video clip now, Smokefree magic.