Automatic Speech Recognition (ASR) in medical contexts has the potential to save time, cut costs, increase report accuracy, and reduce physician burnout. However, the healthcare industry has been slower to adopt this technology, in part due to the importance of avoiding medically-relevant transcription mistakes. In this work, we present the Clinical BERTScore (CBERTScore), an ASR metric that penalizes clinically-relevant mistakes more than others. We demonstrate that this metric more closely aligns with clinician preferences on medical sentences as compared to other metrics (WER, BLUE, METEOR, etc), sometimes by wide margins. We collect a benchmark of 13 clinician preferences on 149 realistic medical sentences called the Clinician Transcript Preference benchmark (CTP), demonstrate that CBERTScore more closely matches what clinicians prefer, and release the benchmark for the community to further develop clinically-aware ASR metrics.