Why don't metals damp vibrations?

Damping implies a loss mechanism. In liquids, where molecules move freely in close proximity, this loss mechanism is a transfer of momentum from one molecule to another.

In pure crystalline metals, the position of atoms in the lattice is fixed, and the forces between them are elastic. That is, if an atom is displaced, it will experience a force that puts it back in its original place, because that place is energetically more favorable. This means the atoms live at the bottom of a potential well. Contrast this with molecules in a liquid which don't have a "favored" place.

As was pointed out by don_gunner94, once you move to "real" metals with a complex microstructure, it is possible to introduce loss mechanisms - such as presented by the graphite in gray cast iron. Under stress, atomic bonds are broken and rearranged; this leads to dissipation of elastic energy, and thus vibration damping. Atoms no longer have a fixed place in the material.

I'd like to point out the example of cast iron. It is renowned for its excellent vibration-damping properties. It is wrong to reach a blanket conclusion saying that metals are bad for vibration damping. The properties of any solid depends part on the material it is made up of and part on the micro-structure of the material. By micro-structure, I mean the arrangement of the grains of that material. Research has led to many new metal alloys being developed that have excellent vibration damping properties. Let me take the example of gray cast iron and talk about its damping properties. If you take a close look at the micro-structure of such irons, you will see a high concentration of graphite in the form of flakes. Higher the amount of graphite, better the vibration damping. But on the other hand, this leads to drastic reduction in the strength of gray cast iron as compared to its other cousins.