Architectures with multiple classes of memory media are becoming a common part of mainstream supercomputer deployments. So called multi-level memories offer differing characteristics for each memory component including variation in bandwidth, latency and capacity. This paper investigates the performance of sparse matrix multiplication kernels on two leading high-performance computing architectures -- Intel's Knights Landing processor and NVIDIA's Pascal GPU. We describe a data placement method and a chunking-based algorithm for our kernels that exploits the existence of the multiple memory spaces in each hardware platform. We evaluate the performance of these methods w.r.t. standard algorithms using the auto-caching mechanisms. Our results show that standard algorithms that exploit cache reuse performed as well as multi-memory-aware algorithms for architectures such as KNLs where the memory subsystems have similar latencies. However, for architectures such as GPUs where memory subsystems differ significantly in both bandwidth and latency, multi-memory-aware methods are crucial for good performance. In addition, our new approaches permit the user to run problems that require larger capacities than the fastest memory of each compute node without depending on the software-managed cache mechanisms.