This work leverages deep learning (DL) techniques in order to do automatic and accurate heart murmur detection from phonocardiogram (PCG) recordings. Two public PCG datasets (CirCor Digiscope 2022 dataset and PCG 2016 dataset) from Physionet online database are utilized to train and test three custom neural networks (NN): a 1D convolutional neural network (CNN), a long short-term memory (LSTM) recurrent neural network (RNN), and a convolutional RNN (C-RNN). Under our proposed method, we first do pre-processing on both datasets in order to prepare the data for the NNs. Key pre-processing steps include the following: denoising, segmentation, re-labeling of noise-only segments, data normalization, and time-frequency analysis of the PCG segments using wavelet scattering transform. To evaluate the performance of the three NNs we have implemented, we conduct four experiments, first three using PCG 2022 dataset, and fourth using PCG 2016 dataset. It turns out that our custom 1D-CNN outperforms other two NNs (LSTM- RNN and C-RNN) as well as the state-of-the-art. Specifically, for experiment E1 (murmur detection using original PCG 2022 dataset), our 1D-CNN model achieves an accuracy of 82.28%, weighted accuracy of 83.81%, F1-score of 65.79%, and and area under receive operating charactertic (AUROC) curve of 90.79%. For experiment E2 (mumur detection using PCG 2022 dataset with unknown class removed), our 1D-CNN model achieves an accuracy of 87.05%, F1-score of 87.72%, and AUROC of 94.4%. For experiment E3 (murmur detection using PCG 2022 dataset with re-labeling of segments), our 1D-CNN model achieves an accuracy of 82.86%, weighted accuracy of 86.30%, F1-score of 81.87%, and AUROC of 93.45%. For experiment E4 (abnormal PCG detection using PCG 2016 dataset), our 1D-CNN model achieves an accuracy of 96.30%, F1-score of 96.29% and AUROC of 98.17%.