This study trained long short-term memory (LSTM) recurrent neural networks (RNNs) incorporating an attention mechanism to predict daily sepsis, myocardial infarction (MI), and vancomycin antibiotic administration over two week patient ICU courses in the MIMIC-III dataset. These models achieved next-day predictive AUC of 0.876 for sepsis, 0.823 for MI, and 0.833 for vancomycin administration. Attention maps built from these models highlighted those times when input variables most influenced predictions and could provide a degree of interpretability to clinicians. These models appeared to attend to variables that were proxies for clinician decision-making, demonstrating a challenge of using flexible deep learning approaches trained with EHR data to build clinical decision support. While continued development and refinement is needed, we believe that such models could one day prove useful in reducing information overload for ICU physicians by providing needed clinical decision support for a variety of clinically important tasks.