  The modified dimension-by-dimension finite volume (FV) WENO method on
Cartesian grids proposed by Buchm\"{u}ller and Helzel can retain the full order
of accuracy of the one-dimensional WENO reconstruction and requires only one
flux computation per interface. The high-order accurate conversion between
face-averaged values and face-center point values is the main ingredient of
this method. In this paper, we derive sixth-order accurate conversion formulas
on three-dimensional Cartesian grids. It is shown that the resulting modified
FV WENO method is efficient and high-order accurate when applied to smooth
nonlinear multidimensional problems, and is robust for calculating non-smooth
nonlinear problems with strong shocks..
