Poetic, Metaphorical, Subtle
Okay, first things first. What makes the writing style of The Eumenides poetic? Here at Shmoop, we use the translation by Christopher Collard in the Oxford World Classics series. We really like this translation a lot, but it actually covers some things up from the poetic standpoint.
As you'll see if you use this translation, the dialogue between the different characters is in prose, whereas the songs sung by the Chorus (plus some of the set-piece songs involving, say, the Chorus, Electra, and Orestes) are in verse, which he makes into a kind of free verse. This is halfway right, except that in the original, absolutely everything is in verse, including the dialogue between characters.
Collard can get away with putting the ordinary dialogue into prose, though, because the poetic meter during the dialogue is a pretty basic one, so called "iambic trimeter," which Aristotle called the one closest to ordinary speech.
What about the songs by the Chorus? There's no question that these are the play's poetic showstoppers. You can also see this in the fact that they use much more high-octane metaphorical language than the rest of the play. Sometimes, these images can be really weird, like when the Chorus of Furies says:
"Such things as these are done by younger gods
with power wholly beyond justice
at the throne dripping with murder
all round its foot, all round its head." (162-168)
Images like these are very cool (and disturbing nightmare food) but hard to put together, and require a lot of imagination. The more imagination and thinking that the language demands from the reader, the more subtle it is.