Going back to Noreen's comment about Irish songs performed in unison, It is interesting to note than when there was a decline in fiddle playing in Sligo, people started "lilting" the old tunes - daidle-eedle-iddle-i-eedle etc.. Hence "diddly diddly" as a term for Irish session tunes. Now, the men and women both sung the tune but in Unison, the men singing one octave lower, obviously. As the fiddle started to re-emerge, the new generation of sligo players started playing in octaves. So, lets assume that Octave unison is the most basic form of harmony, the next most basic is singing in natural fifths - see young tradition's version of Lyke wake dirge, or Cutty Wren. Slightly more advanced is unison at the third - Waterson's soul-cake song uses paralell thirds and fifths - very ancient it sounds too.
Another early harmonic practice is to sing tunes against a burden, or drone, obviously, this creates a harmonic anchor point, so if you have a melody harmonised at the fifth, you can effectively be in two modes at once, the drone is used to give preference to one or the other.
This takes us, in classical harmony terms, up to about 1100 A.D. So I think it's pretty fair to say that folk harmony is not a new idea.