The existence and construction of cycles of various lengths in an interconnection network are important issues in efficiently executing ring-structured parallel algorithms in such a network. The hexagonal honeycomb mesh (HHM) is regarded as a promising candidate for interconnection networks. In this paper we address the problem of how to embed even-length cycles in an HHM. We prove that an HHM of order t ≥ 3 admits a cycle of length l for each even number l such that l = 6 or 10 ≤ l ≤6t - 2. We also describe a systematic method for building these cycles.