We consider a special class of quantum non-dynamical R-matrices in the fundamental representation of GL N with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case N = 2 these are the well-known 6-vertex R-matrix and its 7-vertex deformation. The R-matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type. Namely, we describe the Lax pairs with spectral parameter, the inertia tensors and the Poisson structures. The latter are given by the linear Poisson-Lie brackets for the non-relativistic models, and by the classical Sklyanin type algebras in the relativistic cases. In some particular cases the tops are gauge equivalent to the Calogero-Moser-Sutherland or trigonometric Ruijsenaars-Schneider models.