So far I have got only few constructions with drilled tetracubes. I packed the 6 square tetrominoes with 2 holes in a 2x3x4 box,
which is the only possible construction for a box or rectangle. If we take the 10 square tetrominoes with three holes we can't get any box
at all as an exhaustive search shows.
The 9 L-tetrominoes with one hole can fill a 2x3x6 box and the 36 L-tetrominoes with 2 holes can fill a 2x3x24 box. May be other boxes are
possible and the 84 L-tetrominoes with 3 holes allow for a box, too, but I don't know.