Basically the cube looks like a Rubik's cube, only each of the pieces is not square.

If you start with the center piece (the one you never see in a Rubiks cube), then work your way out from the middle block to the center block on each face, you should end up with a 3-D axis shape which is 15 cm in each direction.  Each axis will consist of 3 blocks and you will be using the 4cm length of one block, the 5cm dimension of the second block, and the 6cm dimesino of the third block.

Once the axis is complete, you can finish off the middle bolc of each edge of the 27cm cube by placing more cubes so that you again get one 4cm dimension, one 5cm dimension, and one 6cm dimension in each of the new 'lines' created.  Make sure you leave one of each dimension for the 8 spaces left on the corners.

If you run into trouble, break down to the original axis shape and rotate one block 90° and continue building from there.

I haven't give the whole solution here, but I think the key to soving this is to start from the center of the cube.