Brownian dynamics simulations are used to study the unraveling process of polymer molecules in dilute solutions under strong elongational flows. We follow chain extension, segmental alignment, and viscosity contribution behavior of individual, randomly coiled, freely jointed bead–rod chain model molecules. In the absence of hydrodynamic shielding, segmental orientation at an intermediate strain rate begins only when aided by overall chain extension. However, at a very high strain rate, rapid initial segmental orientation and lateral chain compression precedes overall chain extension, resulting in the formation of sharp folds in most chains. Fold formation during the extension process is characterized by a sudden decrease in the rate of overall chain extension, an intermediate plateau in birefringence, and a disproportionately low chain end-to-end distance. Hydrodynamic screening generally slows down the uncoiling process, sometimes enough to avoid the formation of folded conformations.