Virtually all microfluidic and nanofluidic devices are fabricated using electrically insulating materials (silicon, glass, or plastics), and their relatively complex features can lead to strongly inhomogeneous electric fields. We will present an approach for simulating DNA electrophoresis in these devices, based on a mixed Boundary Element Method/Brownian Dynamics (BEM/BD) approach. Our method is straightforward and computationally efficient, opening up a new route for simulating DNA electrophoresis in the vast array of devices currently being explored in the microfluidics and nanofluidics community. Moreover, electric field calculations by boundary elements are ideally suited for considering active nanodevices with dynamically changing shapes and electric fields. To demonstrate the accuracy of our method, we studied the collision of a large DNA molecule with an isolated, finite-sized, insulating post. As the electric field distribution for such a geometry is known, it provides an ideal test bed for our BEM/BD method. The BEM/BD simulation is indistinguishable from an equivalent BD simulation that uses the exact solution for the electric field. We will also present an analysis of various approaches aimed at reducing the computational cost of the simulations.