Heart sound signals are vital for the machine-assisted detection of congenital heart disease. However, the performance of diagnostic results is limited by noise during heart sound acquisition. A limitation of existing noise reduction schemes is that the pathological components of the signal are weak, which have the potential to be filtered out with the noise. In this research, a novel approach for classifying heart sounds based on median ensemble empirical mode decomposition (MEEMD), Hurst analysis, improved threshold denoising, and neural networks are presented. In decomposing the heart sound signal into several intrinsic mode functions (IMFs), mode mixing and mode splitting can be effectively suppressed by MEEMD. Hurst analysis is adopted for identifying the noisy content of IMFs. Then, the noise-dominated IMFs are denoised by an improved threshold function. Finally, the noise reduction signal is generated by reconstructing the processed components and the other components. A database of 5000 heart sounds from congenital heart disease and normal volunteers was constructed. The Mel spectral coefficients of the denoised signals were used as input vectors to the convolutional neural network for classification to verify the effectiveness of the preprocessing algorithm. An accuracy of 93.8%, a specificity of 93.1%, and a sensitivity of 94.6% were achieved for classifying the normal cases from abnormal one.