Here is a summary, for my own little brain. Comment and argument welcome.
We have poems from the ancient Chinese, the Middle Eastern peoples, Egyptians, etc. and their musical instruments from tombs, at least back to 2500 BC. The complexity of the Chinese instruments found in tombs indicates an advanced musical culture before 2000 BC. But are these songs still sung? No, certainly not to the ancient tunes. We have Greek and Roman poetry, and literary indication that some of it was sung, but the notations are unknown.

We have Christian liturgical music from about 900 AD on, and ancient music groups make a reasonable job of interpreting the directions and early attempts at notation. Much of this can be considered folk, since the authors are unknown. The earliest dated and author-identified European music is from the 1100-1200 AD period, work by Hildegard von Bingen and Leonin (and possibly others in the Iberian area, studies in progress). Some preserved liturgical chant is a little older, but the chanting would not count as music in the sense we are looking for here.
Peoples of the Ukraine and Georgia also may have liturgical music verifiable to the Middle Ages.
Much of what we know of old Hebrew music is from the Sephardic Jews, who were expelled along with the Islamic people from the Iberian Peninsula by Christians, the last gasp in the 1490s. Much interesting music, but is there anything verifiable before about 1200?

North American pueblo culture goes back to at least 1000 AD; material in some of the chants may be this old, but studies show that there have been shifts in belief and therefore in chants.
We have musical instruments from pre-Christian Central and South America. Several groups, including some native, play these instruments but the music played probably doesn't go back more than a few hundred years at most.
Hindu India, because of the nature of their social and religious beliefs, has been called by some anthropologists the only surviving stone age culture, but I can't comment on the antiquity of any of their songs- ignorance complete here. Change undoubtedly occurred here as well, one only has to look at the Mogul and other influences.

The little animal story brought up by Allen C. (often with a moral) is an example of a whole genre of folk tale that undoubtedly is old, but tunes and meters have changed with time and transmission.

Folk music, I think by definition, changes through time. I would not expect any song to persist unchanged over about 1000 years. Music may persist but transferred to new songs, poems are re-written to fit a situation or the nature of the singer. The 1000 year limit is pretty well verified by the music for which we have data. This is liturgical music. Secular music cannot be verified beyond the 13th century.

By the way, when was the tune attached to the Middle English "Sumer is icumen in"??