When I began learning Latin, I really really enjoyed "Lingua Latin Per Se Illustrada" by Hans Orberg. The book basically uses an immersion/deductive technique to teach Latin rather than simply teaching grammatical rules, in other words it tries to teach Latin as a child would actually pick up their first language.
So instead of teaching all the technicalities of the grammatical rules, it introduces simple forms of the grammatical rules within the text gradually with very simple vocabulary and repeats those simple forms over and over again. I am not saying that this is the best way to teach Latin, and I do think it should be used in conjunction with other more traditional "grammar first" books, but for me it proved invaluable in reading Latin more naturally and increasing my reading pace.
You can see the first few pages of the book here:http://www.amazon.com/Lingua-Latina-Familia-Romana-Latin/dp/1585102385/ref=sr_1_1?s=books&ie=UTF8&qid=1301325061&sr=1-1#reader_1585102385
As you can see the book begins very simple, almost as if it is aimed at a very small child, but the difficulty picks up gradually, and by the end of the book you are almost reading genuine Latin.
I was wondering if there is anything out there that uses a similar approach for Ancient Greek, preferably Koine, but I would be willing to try one for Attic too.