The first unique overtones of a simple vibrating string form a major chord. Are there other simple physical structures that would generate other chord types?
The vibrating string and the air column of typical instruments share the same model which will produce a major chords with the first harmonics : this is called (guess what) an "harmonic system". This means you have to look for anharmonic systems if you want something else as a major chord. Basically anything not typically used for music is anharmonic : a metal plate will have a very different harmonic structure from a vibrating string. Percussion instruments are also a rich field in that regard (with notable exceptions such as the timbals and tablas which are built to have something close to a natural harmonic series). You generally don't get a "chord" though.
A (church) bell is an example of an anharmonic system which yet is tuned. Bells are usually tuned to have a minor third as third partial, producing a minor chord.
See for example http://members.efn.org/~qehn/global/building/bars.htm for some information about tuning aluminium bars. You'll get other sounds from different culture by researching non western instrument tuning such as the gamelan.
Not chord-related per se, but an interesting phenomenon nonetheless:
Tubular Bells (essentially metal pipes), along with some other tuned percussion such as Timpani have an interesting property in that they make our brain fill in a missing harmonic fundamental which isn't actually there.
In tubular bells, modes 4, 5, and 6 appear to determine the strike tone and have frequencies in the ratios 92:112:132, or 81:121:169, which are close enough to the ratios 2:3:4 for the ear to consider them nearly harmonic and to use them as a basis for establishing a virtual pitch
This 'Missing' (or 'Phantom') fundamental phenomenon is an interesting effect whereby we hear an 'implied' frequency suggested by the presence of others. This is probably caused by the brain's ability to extrapolate patterns - e.g. if 200Hz, 300Hz and 400Hz tones are present then the brain may fill in the 'missing' fundamental 100Hz.
In fact, this effect is deliberately exploited in some stops of the Pipe Organ, where the effect is often referred to as a 'resultant'. Having a physical pipe of the required length (e.g. a 64ft bass) is often impractical and to get around it we combine two frequencies and get the brain to fill in the one we actually want.
Here, the desired note is sounded along with a related frequency (often a 5th) which results in the brain hearing the fundamental which is common to both.
This doesn't sound as good as a 'true' stop would, but it's a neat trick that exploits this natural phenomenon to do the job!