Sediment transport occurs when the nondimensional fluid shear stress Theta at the bed surface exceeds a minimum value Theta(c). A large collection of data, known as the Shields curve, shows that Theta(c) is primarily a function of the shear Reynolds number Re-*. It is commonly assumed that Theta > Theta(c) (Re*) occurs when the Re-*-dependent fluid forces are too large to maintain static equilibrium for a typical surface grain. A complimentary approach, which remains relatively unexplored, is to identify Theta(c) (Re-*) as the applied shear stress at which grains cannot stop moving. With respect to grain dynamics, Re-* can be viewed as the viscous time scale for a grain to equilibrate to the fluid flow divided by the typical time for the fluid force to accelerate a grain over the characteristic bed roughness. We performed simulations of granular beds sheared by a model fluid, varying only these two time scales. We find that the critical Shields number Theta(c) (Re-*) obtained from the model mimics the Shields curve and is insensitive to the grain properties, the model fluid flow, and the form of the drag law. Quantitative discrepancies between the model results and the Shields curve are consistent with previous calculations of lift forces at varying Re-*. Grains at low Re-* find more stable configurations than those at high Re-* due to differences in the grain reorganization dynamics. Thus, instead of focusing on mechanical equilibrium of a typical grain at the bed surface, Theta(c) (Re-*) may be better described by the stress at which mobile grains cannot find a stable configuration and stop moving.