Electronically nonadiabatic or non-Born-Oppenheimer (non-BO) chemical processes (photodissociation, charge-transfer, etc.) involve a nonradiative change in the electronic state of the system. Molecular dynamics simulations typically treat nuclei as moving classically on a single adiabatic potential energy surface, and these techniques are not immediately generalizable to non-BO systems due to the inherently quantum mechanical nature of electronic transitions. Here we generalize the concept of a single-surface molecular dynamics trajectory to that of a coupled-surface non-BO trajectory that evolves "semiclassically" under the influence of two or more electronic states and their couplings. Five non-BO trajectory methods are discussed. Next, we summarize the results of a series of systematic studies using a database of accurate quantum mechanical reaction probabilities and internal energy distributions for several six-dimensional model bimolecular scattering collisions. The test set includes three kinds of prototypical nonadiabatic interactions: conical intersections, avoided crossings, and regions of weak coupling. We show that the coherent switching with decay of mixing (CSDM) non-BO trajectory method provides a robust and accurate way to extend molecular dynamics to treat electronically nonadiabatic chemistry for all three kinds of nonadiabatic interactions, and we recommend it for molecular dynamics simulations involving nonradiative electronic state changes.