Two manganese oxides built from MnO6 octahedra arranged in layered (Mg-BUS) and tunnel (Mg-TOD) crystal structures are tested for their performance in Li-ion and Na-ion batteries. The layered Mg-BUS consists of an open layered structure with 2D diffusion pathways for charge-carrying ions, while the Mg-TOD is built from structural tunnels (1D diffusion pathways) with the same height as the interlayer spacing in Mg-BUS. Benefiting from the similar chemical composition and crystal structure dimensions of these two materials, we study the role of diffusion channel geometry in reversible ion intercalation/deintercalation. It was found that in both Li-ion and Na-ion batteries, the two materials have similar initial capacities of ~120-130 mAh g-1. The Mg-TOD demonstrated superior cycling stability in both battery systems, indicating the tunnel structure is advantageous for extended cycling. Rate performance experiments show that in Na-ion batteries, Mg-BUS maintains a higher capacity at higher current rates, suggesting the layered structure allows for more facile diffusion of the larger and heavier Na+ ions. Thus, these results indicate that the tunnel walls, while impeding ion diffusion for the larger Na+ ions, provide structural stability during electrochemical cycling, a finding which can help guide the design of electrode materials for intercalation-based batteries.