We discuss the kinetics of ordering in systems with many degenerate ground states. We make use of the shape of ordered domains which minimizes their interfacial free energy at a given temperature to interpret the dynamical behavior which is observed in Monte Carlo simulations of domain growth. In particular, we argue that the underlying lattice in the simulations can cause different behavior for the kinetics at temperature T=0 versus T0+. This behavior is analyzed in detail in the Potts model and implies that the zero-temperature freezing fixed point in this model is unstable with respect to finite-temperature quenches. We also argue that the choice of boundary conditions can cause subtle finite-size effects which become important much more rapidly in simulations than other sources of finite-size effects.