For metals, whether an electromagnetic (EM) wave is reflected or absorbed is determined primarily by (1) the frequency (or wavelength) of the incident wave, and (2) the density of electrons in the material. This is a consequence of both (a) the response of free electrons to the electric field of EM radiation, and (b) Coulombic forces between these electrons and ionic cores in the metal.
Freely conducting electrons in metals can be collectively be treated as an electron gas whose behavior can actually be described mostly using classical mechanics. In this treatment (see the Lorentz-Oscillator model, a specific example of a driven damped harmonic oscillator), an electron can be accelerated due to a driving force—in this case, the electric field that comprises EM radiation. Other forces affect the electron's motion as well, namely: (1) the Coulombic attraction between electrons and ionic cores within the metal, which provide a restoring force; and (2) damping forces, such as the scattering of electrons off of ion cores.
The combination of all these forces results in a resonant frequency existing for the system. At frequencies below the resonant frequency, input energy is not easily absorbed into the system. (This situation is not unlike a mass on a spring responding to some input force.) Since the energy has to go somewhere, it is instead reflected back outwards. However, at or above the resonant frequency, input energy can be absorbed. For an electron gas in metals, this resonant frequency is called the plasma frequency.
For most metals, the plasma frequency is somewhere in the ultraviolet range, which is far higher in frequency (and thus in energy) than microwaves. This is why microwaves are reflected by sheet metals (the case of mesh grids is different, as they rely on mechanisms that a Faraday cage operates on).
Addendum: there actually is also a quantum-mechanical mechanism for absorption of EM radiation, which is interband absorption—this results from the electronic band structure in all ordered (non-amorphous) materials. Interband absorption explains why some metals are visibly different in appearance (e.g., copper and gold) when these interband transitions occur in the visible region of the EM spectrum. Still, optical properties in most metals are dominated by the position of the plasma frequency rather than interband transitions—hency why most metals have a similar shiny gray/silver reflective appearance.