Predicting the health risks of patients using Electronic Health Records (EHR) has attracted considerable attention in recent years, especially with the development of deep learning techniques. Health risk refers to the probability of the occurrence of a specific health outcome for a specific patient. The predicted risks can be used to support decision-making by healthcare professionals. EHRs are structured patient journey data. Each patient journey contains a chronological set of clinical events, and within each clinical event, there is a set of clinical/medical activities. Due to variations of patient conditions and treatment needs, EHR patient journey data has an inherently high degree of missingness that contains important information affecting relationships among variables, including time. Existing deep learning-based models generate imputed values for missing values when learning the relationships. However, imputed data in EHR patient journey data may distort the clinical meaning of the original EHR patient journey data, resulting in classification bias. This paper proposes a novel end-to-end approach to modeling EHR patient journey data with Integrated Convolutional and Recurrent Neural Networks. Our model can capture both long- and short-term temporal patterns within each patient journey and effectively handle the high degree of missingness in EHR data without any imputation data generation. Extensive experimental results using the proposed model on two real-world datasets demonstrate robust performance as well as superior prediction accuracy compared to existing state-of-the-art imputation-based prediction methods.