Sudoku Weave

Standard Sudoku rules apply, with the following changes: the rows and column have been replaced with the "weaving" diagonals that go down the grid, bouncing at the edges of the grid, as indicated by the black and gray lines. The digits 1 to 9 appear once in each of the following 27 regions: the 9 black diagonal, the 9 gray diagonals, and the 9 3x3 boxes (shaded in white and gray).