One of the most common questions that one faces when first encountering the communicative method of learning Greek is how they themselves can learn in this manner. A number of helpful resources are listed below:
Biblical Language Center (BLC): The seeds of Biblical Language Center (BLC) began more than twenty years ago when Randall Buth, a long-time Bible translator, saw the need for learning the biblical languages communicatively. This led to him moving to Israel in 1996 where he would develop curriculum and materials for learning Biblical Hebrew and eventually Biblical Greek as living languages. Randall Buth is an excellent scholar and all of us in the field of teaching the biblical languages communicatively owe him a debt of gratitude for how he has moved the field along. They run a series of online classes in Living Koine Greek and also sell a number of language-learning materials. They use—in fact, originated—the pronunciation I use on this website.
Polis Institute: The Polis Institute in Jerusalem, founded in 2011, also has one of the best programs in the world for learning Greek communicatively. Christophe Rico, their founder, is exceptional, has a high degree of fluency and has produced some very good materials. My only critique of Polis is that they use a more traditional academic pronunciation and not a restored Koine pronunciation.
LivingText.org: This is currently in development by Jonathon Robie (moderator at B-Greek, see below) and Michael Palmer. It is a free tool that works in a similar way to some of the pedagogy of BLC. Currently only a few lessons are available, but I anticipate more to come in the future.
Koine-Greek.com (with a hyphen!): This is a very helpful blog and collection of resources on Koine Greek. It is run by a number of Greek scholars (primarily Mike Aubrey): Mike Aubrey, Rachel Aubrey, Chris Fresch, Andrew Keenan, and Kris Lyle. They have a lot of helpful resources on Koine Greek linguistics, both from an academic perspective and from a more accessible perspective. In addition, they also conduct a series of interviews with significant scholars in the field. All in all, this is a very helpful website if you are looking to grow in your understanding of the nuts and bolts of Greek and get a feel for the academic field of Koine Greek linguistics as a whole.
GreekLinguistics.com: This website, run by Michael Palmer, contains numerous helpful resources related to Koine Greek linguistics, including a blog, bibliography, and grammatical lessons.
B-Greek: B-Greek is a forum dedicated specifically to the discussion of Biblical/Koine Greek.
Text Kit: Text Kit is a forum dedicated to the wider discussion of the classical languages: Greek and Latin. Within this, there are subforums for discussing Classical Greek and Koine Greek.