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:
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.