It seems that Plato was very fond of the Dorian and Phrygian modes (at least at the time he wrote the Republic), while Confucian put in his ideal government the "music of the Shao and Wu." (Analects 15.11).
Do we have any music from the Shao and Wu? What about the survival of the Dorian and Phrygian modes?
Not quite as old as Plato, but the First Delphic Hymn was written in the Phrygian mode by Limenius in the 2nd century BC, and found inscribed on the Treasury of the Athenians at Delphi.
You might also be interested in the Ensemble Kérylos, which performs surviving music from the Classical period.
References
Pöhlmann and West (2001) Documents of Ancient Greek Music: The Extant Melodies and Fragments.
Very important to note about the (very interesting and very much alive!) field of documenting and preserving ancient music:
As with old language, there are some things in ancient notated music that are left out. The idea that notation should be exact and explicit is quite a modern concept, and even today you will find notated music written not long ago that we actually need recordings to know exactly how it should sound. Notating is a problematic art, and will remain so – if preciseness is wanted, which in many cases it isn't for artistic reasons.
So what does this mean for ancient music? The most common thing is that we have lots of sources on the melodic and harmonic content of ancient pieces, but the rhytm is left out! /u/Karvina mentions the First Delphic Hymn (and Ensemble Kérylos), which have heavy hints to the rhythm in its poetical construction – the text is all that shows when pitches change, pitch notation and lyrics are wreitten together – but we can't be entirely sure about all rhythmic content. For example, should all beats be measured equally, in a somewhat steadfast and quantized way, or should there be some kind of "swing feel" (to sort of misuse a word since it is easily google-able) to it? Is there a freedom of reading, can you hold certain notes or inject rests? These are some of the very basic questions people like the aforementioned Ensemble Kérylos ask in order to render the music. And they do have more answers than me, they are the experts when it comes to ancient Greek music.
So, in essence, we have many sources on music from ancient times, but we do not know enough contextually to give a completely faithful performance. I have mentioned pitch and rhythm, the most common components of music in the common mind, but I have not yet brought up things like ornamentation, timbre or playing techniques, tuning/intonation (easier to know in Greek music, a tad less so in Gregorian singing) and so on. Think especially about voice, how would we ever accurately know the style of people's singing without a recording?