The Sudoku puzzle typically consists of a nine-by-nine grid, in which some of the spaces contain numbers; most of the spaces are blank. The goal is to fill in the blanks with digits from 1 to 9 so that each row, each column, and each of the nine three-by-three blocks making up the grid contains just one of each of the nine digits. Recently Subramani and Ponnuswamy (2009) have considered the Sudoku puzzle as an experimental design and introduced the concept of Sudoku designs. The Sudoku designs are similar to that of latin square designs but accommodate some additional factors. The method of constructing the Sudoku square designs, analysis and applications are also given by Subramani and Ponnuswamy (2009). In this paper we have extended the Sudoku designs to Orthogonal (Graeco) Sudoku square designs in line with that of the Orthogonal (Graeco) latin square designs. A simple method of constructing Graeco Sudoku square designs (GSSD) of odd orders is presented. The proposed method is explained with the help of numerical examples.