In this paper a novel variable selection method based on Radial Basis Function (RBF) neural networks and genetic algorithms is presented. The fuzzy means algorithm is utilized as the training method for the RBF networks, due to its inherent speed, the deterministic approach of selecting the hidden node centers and the fact that it involves only a single tuning parameter. The trade-off between the accuracy and parsimony of the produced model is handled by using Final Prediction Error criterion, based on the RBF training and validation errors, as a fitness function of the proposed genetic algorithm. The tuning parameter required by the fuzzy means algorithm is treated as a free variable by the genetic algorithm. The proposed method was tested in benchmark data sets stemming from the scientific communities of time-series prediction and medicinal chemistry and produced promising results.