A sandwich structure, comprising: an architected core having a first major surface and a second major surface facing away from the first major surface; a first facesheet secured to the first major surface; and a second facesheet secured to the second major surface, the core comprising, at the first major surface, a first plurality of pedestals, each of the first plurality of pedestals being separated from the first facesheet by a base gap of a first plurality of base gaps, the base gap having a width, and being joined to the first facesheet by a compliant base layer of a first plurality of compliant base layers in the first base gap, a thickness of the compliant base layer being equal to the width of the base gap, and a Young's modulus of the compliant base layer being less than a Young's modulus of the first facesheet, wherein a Young's modulus of the compliant base layer being less than 100 MPa; wherein the core comprises, as a major component, a micro-truss material comprising: a plurality of first truss members defined by a plurality of first self-propagating polymer waveguides and extending along a first direction; and the structure further comprising: a first plurality of pedestal casings, each corresponding to a respective one of the first plurality of pedestals, each pedestal casing secured to the first facesheet and surrounding the corresponding pedestal, an interior side wall of each pedestal casing separated from a side wall of the corresponding pedestal by a side gap, of a plurality of side gaps, having a width; and a plurality of compliant side layers, each compliant side layer corresponding to a respective one of the first plurality of pedestals, each compliant side layer having a thickness equal to the width of the side gap and filling the side gap. an architected core having a first major surface and a second major surface facing away from the first major surface; a first facesheet secured to the first major surface; and a second facesheet secured to the second major surface, the core comprising, at the first major surface, a first plurality of pedestals, each of the first plurality of pedestals being separated from the first facesheet by a base gap of a first plurality of base gaps, the base gap having a width, and being joined to the first facesheet by a compliant base layer of a first plurality of compliant base layers in the first base gap, a thickness of the compliant base layer being equal to the width of the base gap, and a Young's modulus of the compliant base layer being less than a Young's modulus of the first facesheet, wherein a Young's modulus of the compliant base layer being less than 100 MPa; being separated from the first facesheet by a base gap of a first plurality of base gaps, the base gap having a width, and being joined to the first facesheet by a compliant base layer of a first plurality of compliant base layers in the first base gap, a thickness of the compliant base layer being equal to the width of the base gap, and a Young's modulus of the compliant base layer being less than a Young's modulus of the first facesheet, wherein a Young's modulus of the compliant base layer being less than 100 MPa; wherein the core comprises, as a major component, a micro-truss material comprising: a plurality of first truss members defined by a plurality of first self-propagating polymer waveguides and extending along a first direction; and the structure further comprising: a first plurality of pedestal casings, each corresponding to a respective one of the first plurality of pedestals, each pedestal casing secured to the first facesheet and surrounding the corresponding pedestal, an interior side wall of each pedestal casing separated from a side wall of the corresponding pedestal by a side gap, of a plurality of side gaps, having a width; and a plurality of compliant side layers, each compliant side layer corresponding to a respective one of the first plurality of pedestals, each compliant side layer having a thickness equal to the width of the side gap and filling the side gap. a first plurality of pedestal casings, each corresponding to a respective one of the first plurality of pedestals, each pedestal casing secured to the first facesheet and surrounding the corresponding pedestal, an interior side wall of each pedestal casing separated from a side wall of the corresponding pedestal by a side gap, of a plurality of side gaps, having a width; and a plurality of compliant side layers, each compliant side layer corresponding to a respective one of the first plurality of pedestals, each compliant side layer having a thickness equal to the width of the side gap and filling the side gap.