In this paper, we propose a deep learning-based psychological stress detection algorithm using speech signals. With increasing demands for communication between human and intelligent systems, automatic stress detection is becoming an interesting research topic. Stress can be reliably detected by measuring the level of specific hormones (e.g., cortisol), but this is not a convenient method for the detection of stress in human-machine interactions. The proposed algorithm first extracts mel-filterbank coefficients using preprocessed speech data and then predicts the status of stress output using a binary decision criterion (i.e., stressed or unstressed) using long short-term memory (LSTM) and feed-forward networks. To evaluate the performance of the proposed algorithm, speech, video, and bio-signal data were collected in a well-controlled environment. We utilized only speech signals in the decision process from subjects whose salivary cortisol level varies over 10%. Using the proposed algorithm, we achieved 66.4% accuracy in detecting the stress state from 25 subjects, thereby demonstrating the possibility of utilizing speech signals for automatic stress detection.