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Machine learning (ML) methods with good applicability to complex and highly nonlinear sequences have been attracting much attention in recent years for predictions of complicated mechanical properties of various materials. As one of the widely known ML methods, back-propagation (BP) neural networks with and without optimization by genetic algorithm (GA) are also established for comparisons of time cost and prediction error. With the aim to further increase the prediction accuracy and efficiency, this paper proposes a long short-term memory (LSTM) networks model to predict the dynamic compressive performance of concrete-like materials at high strain rates. Dynamic explicit analysis is performed in the finite element (FE) software ABAQUS to simulate various waveforms in the split Hopkinson pressure bar (SHPB) experiments by applying different stress waves in the incident bar. The FE simulation accuracy is validated against SHPB experimental results from the viewpoint of dynamic increase factor. In order to cover more extensive loading scenarios, 60 sets of FE simulations are conducted in this paper to generate three kinds of waveforms in the incident and transmission bars of SHPB experiments. By training the proposed three networks, the nonlinear mapping relations can be reasonably established between incident, reflect, and transmission waves. Statistical measures are used to quantify the network prediction accuracy, confirming that the predicted stress-strain curves of concrete-like materials at high strain rates by the proposed networks agree sufficiently with those by FE simulations. It is found that compared with BP network, the GA-BP network can effectively stabilize the network structure, indicating that the GA optimization improves the prediction accuracy of the SHPB dynamic responses by performing the crossover and mutation operations of weights and thresholds in the original BP network. By eliminating the long-time dependencies, the proposed LSTM network achieves better results than the BP and GA-BP networks, since smaller mean square error (MSE) and higher correlation coefficient are achieved. More importantly, the proposed LSTM algorithm, after the training process with a limited number of FE simulations, could replace the time-consuming and laborious FE pre- and post-processing and modelling.