Heavy rain causes the highest drop in travel speeds compared with light and moderate rain because it can easily induce flooding on road surfaces, which can continue to hinder urban transportation even after the rainfall is over. However, very few studies have specialized in researching the multistage impacts of the heavy rain process on urban roads, and the cumulative effects of heavy rain in road networks are often overlooked. In this study, the heavy rain process is divided into three consecutive stages, i.e., prepeak, peak, and postpeak. The impact of heavy rain on a road is represented by a three-dimensional traffic speed change ratio vector. Then, the k-means clustering method is implemented to reveal the distinct patterns of speed change ratio vectors. Finally, the characteristics of the links in each cluster are analyzed. An empirical study of Shenzhen, China suggests that there are three major impact patterns in links. The differences among links associated with the three impact patterns are related to the road category, travel speeds in no rain days, and the number of transportation facilities. The findings in this research can contribute to a more in-depth understanding of the relationship between the heavy rain process and the travel speeds of urban roads and provide valuable information for traffic management and personal travel in heavy rain weather.