Microelectromechanical systems remain an area of significant interest in fundamental and applied research due to their wide ranging applications. Most device designs, however, are largely 2D and constrained to only a few simple geometries. Achieving tunable resonant frequencies or broad operational bandwidths requires complex components and/or fabrication processes. The work presented here reports unusual classes of 3D micromechanical systems in the form of vibratory platforms assembled by controlled compressive buckling. Such 3D structures can be fabricated across a broad range of length scales and from various materials, including soft polymers, monocrystalline silicon, and their composites, resulting in a wide scope of achievable resonant frequencies and mechanical behaviors. Platforms designed with multistable mechanical responses and vibrationally decoupled constituent elements offer improved bandwidth and frequency tunability. Furthermore, the resonant frequencies can be controlled through deformations of an underlying elastomeric substrate. Systematic experimental and computational studies include structures with diverse geometries, ranging from tables, cages, rings, ring-crosses, ring-disks, two-floor ribbons, flowers, umbrellas, triple-cantilever platforms, and asymmetric circular helices, to multilayer constructions. These ideas form the foundations for engineering designs that complement those supported by conventional, micro-electromechanical systems, with capabilities that could be useful in systems for biosensing, energy harvesting, and others.