Evaluations of multiple environment trials (MET) often reveal substantial genotype by environment interactions, and the effects of genotypes within environments are often estimated using cell means, i.e. the simple mean of the observations of each genotype in each environment.  However, these estimates are inaccurate, especially for unreplicated or partially replicated trials, so alternative methods of analysis are necessary.  One possible approach utilizes information, often from pedigree data, about relationships among the tested genotypes through the use of a genetic relationship matrix (GRM).  Predictive accuracy may also be improved by the use of factor analytic (FA) structures for environmental covariances.  In this study, data were simulated to resemble results from a range of MET.  These simulated data sets covered a range of scenarios with varying numbers of environments and genotypes, environmental relationship patterns, field trial designs, and magnitudes of experimental error.  The simulated data were used to evaluate 20 mixed models, ten of which included GRMs and ten which did not.  The models included ten structures for environmental covariances including structures with no environmental correlation, structures with constant correlation among environments, and six FA structures.  These models were compared to each other and to cell means and Additive Main effects and Multiplicative Interaction (AMMI) methods in terms of successful convergence and predictive accuracy.  For most of the scenarios, models which included a GRM and a compound symmetric, constant variance structure produced the most accurate estimates.  Models with GRM and FA structures were more accurate only when used to evaluate scenarios simulated with Toeplitz patterns of relationships and more than 25 genotypes or five environments.  Unfortunately, the improved accuracy with the FA structures in these scenarios came at the cost of reduced convergence rates, so FA structures may not be reliable enough for some uses.