We have performed cough detection based on measurements from an accelerometer attached to the patient's bed. This form of monitoring is less intrusive than body-attached accelerometer sensors, and sidesteps privacy concerns encountered when using audio for cough detection. For our experiments, we have compiled a manually-annotated dataset containing the acceleration signals of approximately 6000 cough and 68000 non-cough events from 14 adult male patients in a tuberculosis clinic. As classifiers, we have considered convolutional neural networks (CNN), long-short-term-memory (LSTM) networks, and a residual neural network (Resnet50). We find that all classifiers are able to distinguish between the acceleration signals due to coughing and those due to other activities including sneezing, throat-clearing and movement in the bed with high accuracy. The Resnet50 performs the best, achieving an area under the ROC curve (AUC) exceeding 0.98 in cross-validation experiments. We conclude that high-accuracy cough monitoring based only on measurements from the accelerometer in a consumer smartphone is possible. Since the need to gather audio is avoided and therefore privacy is inherently protected, and since the accelerometer is attached to the bed and not worn, this form of monitoring may represent a more convenient and readily accepted method of long-term patient cough monitoring.