You can study Hellenistic Greek, the form of the Greek language found in the Christian New Testament, the writings of Stoic philosophers such as Epictetus, and a wealth of other documents from the period of the Greek and Roman empires, at our sister site, HellenisticGreek.com.
The beginnings of an approach incorporating elements of a communicative model to learning Hellenistic Greek can be found at HellenisticGreek.com. These materials cannot replace a face to face class, but can be used free of charge either as a part of such a class or on their own. The materials are far from complete, but can give you a taste of learning Greek in Greek.
There are a few websites that allow you to purchase materials for a communicative approach or take a tuition based class as well as one recent addition that provides videos exhibiting a communicative approach that is completely free.
Free Greek Online
For open access to materials to acquire Hellenistic Greek, few things can match the contribution of FreeGreek.online through the series of videos, Alpha with Angela. The methodology is sound, and lessons are very well thought out.
The pronunciation used in Alpha with Angela does not represent the way Greek sounded in the hellenistic period. Following the example of the Polis Institute (discussed breifly below), the videos use a pronunciation system that supports learning Hellenistic Greek spelling rather than the authentic reconstructed pronunciation of the period.
Randall Buth
In 1996 Randall Buth moved to Israel and began developing new curriculum and classroom methodologies for teaching biblical languages. He was the first to adapt methods for learning a modern language to the study of biblical languages. The results of this effort can be found at the Biblical Language Center. Dr. Buth is a widely recognized voice in the field of communicative approaches to Biblical Greek.
This course requires a trip to Israel.
Michael Halcomb
Paul Nitz
Paul Nitz teaches koine Greek in Malawi using communicative methods. Many of his lessons are available free of charge on Youtube. His approach is different from the two above in that he uses the traditional Erasmian pronunciation rather than Reconstructed Koine.
The number of websites offering a traditional approach to teaching Koine Greek is too great to list. Many require you to buy a textbook or pay a fee to use the online materials. I list one site here because it is completely free.
Corey Keating
Corey Keating has had a set of lessons for learning New Testament Greek online for quite some time now. His materials appear to have last been updated in 2010.
Christophe Rico
Christophe Rico teaches at Polis Institute in Israel (Polis: the Jerusalem Institute of Languages and Humanities). He uses a communicative method, but does not use Reconstructed Attic or Reconstructed Koine pronunciation. He used the traditional Erasmean system instead.
The Polis Institute does much more than teach Ancient Greek. In all language classes, regardless of the language, the Institute uses what it calls the "Polis Method." You can read about this method at the institute's website.
Donald J. Mastronarde
With the assistance of the Berkeley Language Center of the University of California, Berkeley, Donald J. Mastronarde has made available a wonderful set of online Ancient Greek Tutorials based on his book, Introduction to Attic Greek. While these tutorials are based on Mastronarde's book, most of them are very usable without it!
Jeffrey A. Rydberg-Cox
For more advanced students, Jeffrey A. Rydberg-Cox provides an overview of Greek syntax based on Herbert Wier Smyth's A Greek Grammar for Colleges.
Jonathan Slocum
Jonathan Slocum of the University of Texas at Austin has edited and updated a series of lessons for Classical Greek originally produced with Winfred P. Lehmann.
Textkit
Textkit provides downloadable copies of a wonderful variety of Classical Greek textbooks on their Greek and Latin Learning Tools page.
Elaine Woodward and Marianne Pagos
Elaine Woodward and Marianne Pagos, teachers at Boston Latin School, have provided a concise handbook (the Greek Enchiridion) to assist you as you learn to read Classical Greek.
Check out the Readers Page to find downloadable Greek texts designed to help you improve your reading ability.