Yes, Greek Accents Matter (And Elisions Weren't Entirely Elided)

In 408 BC, the Grecian tragic actor Hegelokhos misspoke while playing the eponymous protagonist during the premier performance of Euripides' Orestes, and caused something of a stir which would be remembered for years afterwards by the Athenians and for millenia by classicists as a cautionary tale told to Greek students to watch their pronunciation.

Line 279 of the tragedy runs as: "Ἐκ κυμάτων γὰρ αὖθις αὖ γαλήν' ὁρῶ. - For out of the storms I again a stillness see," and the poor actor ran out of breath and incorrectly pronounced the accent of the word "calm" (γαλήν', elided from its full form γαληνά) as a circumflex closing the final syllable instead of an acute preceding the missing final syllable, so that it sounded as though he said "weasel, stoat, polecat" (γαλῆν, from γαλῆ): "For out of the storms I again a stoat do see."

The Athenians, who were rather touchy about their tragedy, laughed, booed, and hissed, thus turning a rather innocuous and throw-away line from Euripides into a font of comedic lampooning.
Contemporary comic playwrights included references to seeing storms and stoats for years to come, such as when Dionysos' clever companion Xanthias attempts to cheer up the unhappy god bewailing his fate in Aristophanes' comedy, Frogs:

Ξανθίας
θάρρει: πάντ᾽ ἀγαθὰ πεπράγαμεν,
            Cheer up! Everything we have done is right!
ἔξεστί θ᾽ ὥσπερ Ἡγέλοχος ἡμῖν λέγειν,
            And, just like Hegelokhos, we can say:
‘ἐκ κυμάτων γὰρ αὖθις αὖ γαλῆν ὁρῶ.’
            "For out of storms I again a stoat do see!"

                                                   -Aristophanes, Frogs 302-304. Trans is my own.

Xanthias is pronouncing the misspoken line as if it were a gnomic saying, a proverb, or wise words of comfort. 

A lost comedy (fragments survive) of the playwright Strattis contains this gem:

A. γαλῆν᾿ ὁρῶ.
           Speaker A: I see a stoat!
Β. ποῖ, πρὸς θεῶν, ποῖ ποῖ γαλῆν;
           Speaker B: Where? By the gods, where is the stoat?
Α. γαληνά.
           Speaker A: No, "stillness".
Β. ἐγὼ δ᾿ ᾤμην σε “γαλῆν” λέγειν “ὁρῶ.”
           Speaker B: Huh, I thought you said, "I see a stoat!" 

                                                   -Strattis Fragment 63. Trans. is my own.

So yes, "those marky thingies above those weird Greek letters" actually matter, and elided syllables were not completely lost and left un-pronounced by native speakers of the Classical Age.