A generalized two-dimensional word is a function on Z² with a finite number of values. The main problem we are interested in is periodicity of two-dimensional words satisfying some local conditions. Let A be a matrix of order n. The function φ : Z² → Rⁿ is a generalized centered function of radius r with the matrix A ifunder(∑, y ∈ Z² : 0 < | y - x | ≤ r) - 35 pt φ (y) = φ (x) Afor every x ∈ Z², where for x = (x₁, x₂), y = (y₁, y₂) we have | y - x | = | y₁ - x₁ | + | y₂ - x₂ |. We prove that every generalized centered function of radius r > 1 with a finite number of values is periodic. For r = 1 the existence of non-periodic generalized centered functions depends on the spectrum of the matrix A. Similar results are obtained for the infinite triangular and hexagonal grids.