The ability to predict complex traits from marker data is becoming increasingly important in plant and animal breeding. One approach is to first identify significant markers and then build a multiple regression model.  A more recent strategy has been to include all markers using the computational method ridge regression-BLUP (RR-BLUP).   To capture the best of both approaches in a non-Bayesian framework, a new prediction method has been developed that couples RR-BLUP with genome-wide association analysis.  The algorithm, called RR-BLUP+MR (Marker Reduction), is available as part of a new software package for R.  A core utility in the package is a fast maximum likelihood algorithm for solving RR-BLUP mixed models.  This function is used both for association analysis and to identify the ridge parameter, making the software applicable to unreplicated data.  The speed and prediction accuracy of RR-BLUP+MR were compared against RR-BLUP and Bayesian LASSO using structured maize and wheat populations.  For grain yield, flowering time, and simulated phenotypes with over 25 QTL, the three methods had similar accuracies.  For simulated traits with less than 25 QTL, the accuracy of RR-BLUP+MR was significantly greater.  These results demonstrate that RR-BLUP+MR performs well across the full range of genetic complexities.