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One way to increase the ductility of metallic glasses is to induce heterogeneous microstructures, as for example in nanoglasses and crystal-glass composites. The heterogeneities have important consequences not only on the development of shear bands, but also on the structural relaxations in the glass phase. Experiments using dynamic mechanical spectroscopy (DMS) have been conducted, but an atomic-scale picture is still lacking. Here we apply DMS within molecular dynamics simulations to a classical CuZr metallic glass to study how structural relaxations and mechanical energy dissipation are affected by different microstructures, including nanoglasses, crystal-glass nanolaminates and glasses with spherical crystalline inclusions. We find that in a fully glassy system, not only the fraction but also the spatial homogeneity of "hard" icosahedral environments matter. When hard crystalline particles are introduced, the storage modulus simply results from volumetric averages consistent with the classical Voigt and Reuss bounds. On the other hand, loss moduli are much more complex and can be smaller or larger than in a pure glass depending on the microstructure and loading condition. Atomistic processes leading to these evolutions are discussed and remaining open questions are highlighted. [Display omitted] [ABSTRACT FROM AUTHOR]