Detailed knowledge of the structure and growth mechanism of amyloid fibrils is important for understanding the disease process. Recently, solid-state NMR and other spectroscopic data have revealed the equilibrium organization of the tertiary structure of fibrils formed by various segments of beta-amyloid peptides. A three-step "dock-and-lock" mechanism for fibril growth has been proposed on the basis of the kinetic data. Here we use all-atom replica-exchange molecular dynamics simulations in generalized-Born implicit solvent to probe the mechanism of tertiary structure propagation in fibrils of Abeta(16-22) modeled as an oligomer consisting of two beta-sheets each having four strands. The data show that following association with the oligomer, but before being fully locked onto the existing beta-sheet, the added monomer predominantly samples states with the antiparallel strand orientation, but both in- and one-residue shifted backbone hydrogen bond alignments. The in-register state, which is the experimentally observed equilibrium alignment, is marginally more stable than the registry-shifted one. These results suggest that, following the fast docking step, the added monomer dynamically slides in the backbone registry, and stabilization of the preferential alignment must occur in the second locking step as the monomer becomes fully integrated with the fibril. We also delineate the electrostatic and hydrophobic effects in directing the registry alignment during monomer addition. Surprisingly, the in-register alignment provides both increased cross-strand electrostatic attraction and hydrophobic surface burial. Finally, our data support the notion that side chain hydrophobic burial is a major driving force for beta-sheet assembly.