Recently there was found a new method to compute the (generalized) Poincare series of some multi-index filtrations on rings of functions. First the authors had elaborated it for computing the Poincare series of the filtration on the ring OC2,0 of germs of functions of two variables defined by irreducible components of a plane curve singularity. The corresponding formula (announced in [2]) was proved in [3] by another method. The new method uses the notion of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions. This notion is similar to (and inspired by) the notion of the motivic integration. Here we apply this method for computing the Poincare series of a (natural) multi-index filtration on the ring of germs of functions on a rational surface singularity. We explicitly calculate the coefficients of this series. Let (S, 0) be a germ of an (isolated) rational surface singularity and let π : (X,D) → (S, 0) be its minimal resolution. Here X is a smooth surface, π is a proper analytic map which is an isomorphism outside of D = π−1(0), and the exceptional divisor D is the union of irreducible components Ei (i = 1, . . . , r) transversal to each other, each component Ei is isomorphic to the projective line CP1.