[en] We study harmonic polynomials on the quantum Euclidean space E^N_q generated by quantum coordinates x_i, i = 1, 2, ..., N, on which the quantum group SO_q(N) acts. They are defined as solutions of the equation Δ_qp = 0, where Δ_q is the q-Laplace operator on E^N_q. We construct a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SO_q(N - 2). The associated spherical polynomials constitute an orthogonal basis of the spaces of homogeneous harmonic polynomials. They are represented as products of polynomials depending on q-radii and x_j, x_j', j' = N - j + 1. This representation is, in fact, a q-analogue of the classical separation of variables