Replica exchange molecular dynamics (REMD) has become a valuable tool in studying complex biomolecular systems. However, its application on distributed computing grids is limited by the heterogeneity of this environment. In this study, we propose a REMD implementation referred to as greedy REMD (gREMD) suitable for computations on heterogeneous grids. To decentralize replica management, gREMD utilizes a precomputed schedule of exchange attempts between temperatures. Our comparison of gREMD against standard REMD suggests four main conclusions. First, gREMD accelerates grid REMD simulations by as much as 40 %. Second, gREMD increases CPU utilization rates in grid REMD by up to 60 %. Third, we argue that gREMD is expected to maintain approximately constant CPU utilization rates and simulation wall-clock times with the increase in the number of replicas. Finally, we show that gREMD correctly implements the REMD algorithm and reproduces the conformational ensemble of a short peptide sampled in our previous standard REMD simulations. We believe that gREMD can find its place in large-scale REMD simulations on heterogeneous computing grids.