Interestingly something sounding metallic has its origin in its dimensions rather than the material, and this effect is only amplified by the material.
The character of perceived sound is mostly determined by the excited oscillations of an object, the so called modal structure. The modes of an object depend on the geometry. An approximately one-dimensional object like a long thin cylinder will have mode frequencies that come as integer multiples of a fundamental frequency. These can be easily grouped by our auditory cortex to result in a single tonal perception with a timbral character depending on the energy distribution in the vibrational modes. We call this frequency structure "harmonic".
A 2 or 3 dimensional object has additional frequencies that cannot be described as harmonic overtones of a fundamental. Often our brain is still able to identify an approximately harmonic structure in the collection of frequencies, especially if the energy distribution in the modes favors almost integer related modes, and we hear a sound that is not quite harmonic, but still has an audible fundamental frequency. This is the typical bell sound. Some sound cleaner, others relatively harsh, depending on how well the harmonic overtone pattern is approximated.
If there is no good harmonic approximation and a lot of modes are excited with high energy then the sound is not tonal anymore. A typical example for this is a cymbal. Now this is all true for wooden objects as well as for metal objects. So what does it have to do with metal? In order for us to be able to hear the modal structure, the modes must decay slowly and they also have to have the right frequency spacing to be discriminated by our ears. Metal happens to have the right combination of speed of sound and decay times to make the modal structure stand out. In combination with the typical shape of metal objects (which is, more than 1-dimensional) you get the inharmonic sound that is often described as "metallic".
Edit: In addition, the stiffness of the metallic object makes high frequency excitations easier, so it's also the energy in the high frequency modes, where the modal density is quite large, that contributes to the metallic sound.