We study modules over the ring ₀ of differential operators with power series coefficients. For ₀-modules, we introduce a new notion of F-Gröbner basis and present an algorithmic method to compute it. Our method is more algebraic than that of Castro (1986, 1987) which is based on the Weierstrass-Hironaka division theorem. The essential point of our method consists in using a filtration of ₀ introduced by Kashiwara (1983). This enables us to extend some of the algorithmic methods for rings of power series to ₀-modules. As applications, we can compute, in some cases, the characteristic variety, and the dimension of the space of solutions, of a system of linear partial differential equations via F-Gröbner bases. The relation to previously known methods is also stated.