I think aloud, and make a (fairly trivial) observation.

The total number of "diagonals", using our definition of the word is 34, that is, 17 in each direction. (There are 8 "shorter" diagonals either side of the "longest" diagonal in each direction; 2 x 8 + 1 = 17). But 12 of these diagonals are irrelevant. These 12 are the 4 groups of 3 nearest each corner: each 3x3 box at a corner of the puzle contains 3 of our diagonals. Now since these diagonals are within a box, the grid satisfying the "normal" su doku properties implies that it will satisfy the property we desire for these 12 diagonals too (because no digit is repeated within the box, so no digit will be repeated within a diagonal). So one way of looking at it is that we only need consider the 22 diagonals (11 in each direction) that aren't wholly within a box.