General multivariate periodic wavelets are an efficient tool for the approximation of multidimensional functions, which feature dominant directions of the periodicity. One-dimensional shift invariant spaces and tensor-product wavelets are generalized to multivariate shift invariant spaces on non-tensor-product patterns. In particular, the algebraic properties of the automorphism group are investigated. Possible patterns are classified. By divisibility considerations, decompositions of shift invariant spaces are given. The results are applied to construct multivariate orthogonal Dirichlet kernels and the respective wavelets. Furthermore a closure theorem is proven.