In this paper we present a new approach for calculating viscoelastic flows. The polymer stress is not determined from a closed-form constitutive equation, but from a microscopic model. In this description, we replace the collection of individual polymer molecules by an ensemble of configuration fields, representing the internal degrees of freedom of the polymers. Similar to the motion of real molecules, these configuration fields are convected and deformed by the flow and are subjected to Brownian motion. We incorporated this field description in a finite element calculation. An important advantage of our approach is that the difficulties associated with particle tracking of individual molecules are circumvented. In order to validate our approach and to demonstrate its robustness, we present the results for the start-up of planar flow of an Oldroyd-B fluid past a cylinder between two parallel plates. The results are very promising. We find excellent agreement between the results of the configuration field formulation and those obtained using a closed-form constitutive equation. Moreover, the microscopic method appears to be more robust than the conventional macroscopic technique.