We tackle the challenging problem of single-source domain generalization (DG) for medical image segmentation, where we train a network on one domain (e.g., CT) and directly apply it to a different domain (e.g., MR) without adapting the model and without requiring images or annotations from the new domain during training. Our method diversifies the source domain through semantic-aware random convolution, where different regions of a source image are augmented differently at training-time, based on their annotation labels. At test-time, we complement the randomization of the training domain via mapping the intensity of target domain images, making them similar to source domain data. We perform a comprehensive evaluation on a variety of cross-modality and cross-center generalization settings for abdominal, whole-heart and prostate segmentation, where we outperform previous DG techniques in a vast majority of experiments. Additionally, we also investigate our method when training on whole-heart CT or MR data and testing on the diastolic and systolic phase of cine MR data captured with different scanner hardware. Overall, our evaluation shows that our method achieves new state-of-the-art performance in DG for medical image segmentation, even matching the performance of the in-domain baseline in several settings.