[iplhmdcf] Filling space with integer bricks

Fill space with a tessellation of 2x1x1 bricks.  For all possible slices, what is the minimum number of bricks you would have to cut through?  This sort of defines the strength of the crystal along that shear plane.  One would only need to consider slices on orthogonal integer planes.  What tessellation is strongest?

A combination of cubes of width 2 and 1?  Inspired by such a 2D tessellation of squares.