This paper finds a way to extend the well-known Fourier methods, to so-called n+1directions partition domains in n-dimension. In particular, in 2-D and 3-D cases, westudy Fourier methods over 3-direction parallel hexagon partitions and 4-direction parallelparallelogram dodecahedron partitions, respectively. It has pointed that, the most conceptsand results of Fourier methods on tensor-product case, such as periodicity, orthogonality ofFourier basis system, partial sum of Fourier series and its approximation behavior, can bemoved on the new non tensor-product partition case.