To employ a Convolutional Neural Network (CNN) in an energy-constrained embedded system, it is critical for the CNN implementation to be highly energy efficient. Many recent studies propose CNN accelerator architectures with custom computation units that try to improve energy-efficiency and performance of CNNs by minimizing data transfers from DRAM-based main memory. However, in these architectures, DRAM is still responsible for half of the overall energy consumption of the system, on average. A key factor of the high energy consumption of DRAM is the refresh overhead, which is estimated to consume 40% of the total DRAM energy. In this paper, we propose a new mechanism, Refresh Triggered Computation (RTC), that exploits the memory access patterns of CNN applications to reduce the number of refresh operations. We propose three RTC designs (min-RTC, mid-RTC, and full-RTC), each of which requires a different level of aggressiveness in terms of customization to the DRAM subsystem. All of our designs have small overhead. Even the most aggressive RTC design (i.e., full-RTC) imposes an area overhead of only 0.18% in a 16 Gb DRAM chip and can have less overhead for denser chips. Our experimental evaluation on six well-known CNNs show that RTC reduces average DRAM energy consumption by 24.4% and 61.3%, for the least aggressive and the most aggressive RTC implementations, respectively. Besides CNNs, we also evaluate our RTC mechanism on three workloads from other domains. We show that RTC saves 31.9% and 16.9% DRAM energy for Face Recognition and Bayesian Confidence Propagation Neural Network (BCPNN), respectively. We believe RTC can be applied to other applications whose memory access patterns remain predictable for a sufficiently long time.