In this paper we establish a new efficient method for simulating polymer-solvent systems which combines a lattice Boltzmann approach for the fluid with a continuum molecular-dynamics (MD) model for the polymer chain. The two parts are coupled by a simple dissipative force while the system is driven by stochastic forces added to both the fluid and the polymer. Extensive tests of the new method for the case of a single polymer chain in a solvent are performed. The dynamic and static scaling properties predicted by analytical theory are validated. In this context, the influence of the finite size of the simulation box is discussed. While usually the finite size corrections scale as L-1 (L denoting the linear dimension of the box), the decay rate of the Rouse modes is only subject to an L-3 finite size effect. Furthermore, the mapping to an existing MD simulation of the same system is done so that all physical input values for the new method can be derived from pure MD simulation. Both methods can thus be compared quantitatively, showing that the new method allows for much larger time steps. Comparison of the results for both methods indicates systematic deviations due to nonperfect match of the static chain conformations.