The modern modes that are playable with the context of the well tempered 12 tone system are named using the ancient Greek appelations, and can be described as the diatonic (piano white key) sequences:

           Ionian:       C D E F G A B (C)
           Dorian:       D E F G A B C (D)
           Phrygian:     E F G A B C D (E)
           Lydian:       F G A B C D E (F)
           Mixolydian:   G A B C D E F (G)
           Aeolian:      A B C D E F G (A)
           Locrian:      B C D E F G A (B)

           Ionian:       T T S T T T S
           Dorian:       T S T T T S T
           Phrygian:     S T T T S T T
           Lydian:       T T T S T T S
           Mixolydian:   T T S T T S T
           Aeolian:      T S T T S T T
           Locrian:      S T T S T T T

One would think that after having corrected the correspondences to the liturgical modes, that then the objects and names would be in accordance with those of the ancient Greeks. One would be even more wrong. So, clearly these are assumptions definitely not to be made.

We are very fortunate to have some rather early theoretical writings by Greeks concerning music. The earliest is seemingly the "Harmonics" of Aristoxenus (ca. 330 BCE), and so we know what the theorists say. Understanding what they mean, however, is an entirely different matter, since written examples of any music to which the written theory may pertain is almost nonexistent.

This ignorance of the reality of ancient Greek music goes so far that we are not quite sure what a Greek mode really was. We only understand ancient Greek music on the level of pitch tempering. Trying to understand or retrodict what the music was like is equivalent to predicting the "Variations and Double Fugue on a Theme of Bach", by Max Reger, merely from a knowledge of well tempering and possibly a bit about chords and classical chromatic harmonic cadential formulas.
It can't be done.

For an interesting and thoughtful essay on the problem that is based in actual research, see The Ancient Musical Modes: What Were They?, together with more links there. The article is by Ed Friedlander, one of the world's good guys who is also as completely interesting in other than his chosen field of pathology as he is in pathology.

From the writings of Aristoxenus, one can deduce that the ancient Greeks used microtonal embellishments, probably not unlike those stylistic practices in modern Greek music, and indeed through out middle eastern cultures. These, unfortunately, have no notational place in western scoring language.

To the Greeks, all thinking about anything serious was philosophy, and writings on music theory were therefore in that culture, philosophy. Ancient Greek philosophers were in one important respect no different that philosophers in any other culture or time: the writings were predominantly prescriptive and proscriptive. See Plato's "Republic", Hobbes "Leviathan" [shudder], or any other poison of choice.

Whether or not the theories of Greek music were actually paid attention to in practice is at very least in question.

It is reasonably clear that all theorizing about music is fundamentally derived from Pythagoras who understood the intervallic concepts that we call the octave, fifth, and fourth. Greek music seems to have been automatically wedded to language either by recitation or by song, but most popularly involved also two instruments, the aulos and the kithara. For some other musical definitions Eric's Treasure Trove: music may be helpful.

If ancient Greek music is so heavily dependent on the sound of language (prosody) then we may learn more about it by knowing and understanding of the sound of (probably Attic) ancient Greek. That is another problem altogether, for scholars of ancient languages.