The use of additive manufacturing allows the production of complex designs, including metallic lattice structures, which combine lightness and good mechanical properties. Herein, the production of AlSi10Mg lattice structures by laser powder bed fusion is explored and consistent processing parameters are selected. Thereafter, it is demonstrated that heat treatments, specifically designed on the basis of the alloy's metallurgy, can be used to selectively induce different microstructural modifications and, consequently, finely tune the static and dynamic mechanical behavior of the aluminum lattice structures. In particular, removal of residual stresses results to be the dominant factor in allowing a smooth quasistatic compressive behavior and improving the structure's ability to absorb energy during collapse. On the contrary, the increase in ductility connected to the spheroidization of the Si network is shown to be of paramount importance in improving the structure's dynamic damping ability by allowing local plasticization.