Listverse released a list of ten ancient Hellenic writers you should know. It's a good list, and the reasoning why these were chosen is a great read in and of itself. The ten who made the cut are:

10. Hómēros -- writer of the Iliad and the Odysseia
9. Sophocles -- the tragedian who wrote (amongst others) Antigone, Oedipus the King and Electra
8. Herodotos -- whose book The Histories is considered the first work of history in Western literature
7. Euripides -- the tragedian who wrote (amongst others) Alcestis, Medea and The Bacchus
6. Hippokrátēs -- who fathered modern medicine, mostly with his Hippocratic Corpus
5. Aristophanes -- the comic playwright who wrote (amongst others) The Clouds, The Wasps and Lysistrata
4. Plato -- one of the fathers of Western philosophy
3. Aristotle -- the last of the great Hellenic philosophers 
2. Euclid -- a mathematician and the father of geometry, whose main work--The Elements--is still used as a textbook in mathematics 
1. Archimedes -- a mathematician, engineer, inventor, physicist and astronomer whose ideas are as the basics of all these disciplines

I most certainly concur that this list lists some of ancient Hellas' most influential writers, but the list is far from complete. Today, I want to add five more to it, and most of my picks will no surprise you.

5. Sappho
Sappho (Σαπφώ) was a Hellenic lyric poet, born on the island of Lesbos (Λέσβος) around 620 BC, although the exact date is unknown. She wrote beautiful and highly romantic poetry that comes and goes straight to the heart. One of the most famous of her works is her Hymn to Aphrodite.

4. Aeschylos
As the original list includes both Sophocles and Euripides, it seems only fair to include Aeschylos. Aeschylos (Aiskhulos, Αἰσχύλος) was the first of the three Hellenic tragedians whose plays can still be read or performed. According to Aristotle, he expanded the number of characters in plays to allow for conflict amongst them, whereas previously characters had interacted only with the chorus. Aescholos' most famous works are undoubtedly the Seven against Thebes, the Supplicants and the Orestia. 

3. Pythagoras
Pythagoras of Samos ( Πυθαγόρας ὁ Σάμιος) was an Ionian Hellenic philosopher, mathematician, and the father of the religious movement called Pythagoreanism. He lived from about 570 BC to about 495 BC, and made influential contributions to philosophy, religious teaching, math, ethics, and science. His most famous work is, undoubtedly, the Pythagorean theorem (a^2 + b^2 = c^2), that's standard in every math textbook. Interestingly enough, not a single bit of writing has been preserved--if he ever wrote anything down to begin with. His works are mostly quoted by his students, or known through critiques by Aristotle.

2. Plutarch
Plutach (Ploútarkhos, Πλούταρχος) was an ancient Hellenic historian, biographer, and essayist who lived between 46 and 120 AD. He is known primarily for his Parallel Lives and Moralia, but much of his work is lost to us. Plutarch's writings are full of details about people and places, and are therefor a true treasure trove.

1. Hesiod
Hesiod (Hesiodos, Ἡσίοδος) was a Hellenic oral poet who lived between 750 and 650 BC, around the same time as Hómēros. I would dare say that his work--especially 'Works and Days' and the 'Theogony'--has shaped the way scholars and practitioners of Hellenismos view ancient Hellenic society, religion and way of life. This is why I strongly feel that anyone who feels drawn to the Theoi, might benefit from investing some time in reading his words.