We perform molecular dynamics (MD) simulations of the crystallization process in binary Lennard-Jones systems during heating and cooling to investigate atomic-scale crystallization kinetics in glass-forming materials. For the cooling protocol, we prepared equilibrated liquids above the liquidus temperature Tl and cooled each sample to zero temperature at rate R-c. For the heating protocol, we first cooled equilibrated liquids to zero temperature at rate R-p and then heated the samples to temperature T > T-l at rate R-h. We measured the critical heating and cooling rates R-h(*) and R-c(*) , below which the systems begin to form a substantial fraction of crystalline clusters during the heating and cooling protocols. We show that R-h(*) > R-c(*) and that the asymmetry ratio R-h(*) / R-c(*) includes an intrinsic contribution that increases with the glass-forming ability (GFA) of the system and a preparation-rate dependent contribution that increases strongly as R-p -> R-c(*) from above. We also show that the predictions from classical nucleation theory (CNT) can qualitatively describe the dependence of the asymmetry ratio on the GFA and preparation rate Rp from the MD simulations and results for the asymmetry ratio measured in Zr-and Au-based bulk metallic glasses (BMG). This work emphasizes the need for and benefits of an improved understanding of crystallization processes in BMGs and other glass-forming systems.