I came across Hans Orberg's excellent Lingua Latina books some time ago. He teaches Latin by using only Latin, accompanied by little pictures along with grammatical notes and tables. You read Latin that makes sense from the first page (and only Latin), and I found his approach truly excellent.

But now I find that my preference is for Greek works, and I would like to be able to make similar progress in Greek. Does anyone know of a book that teaches Greek using the method we find in Orberg's books?

There is a great book, written for the first three classes of high school children in Greece, (by the Athens College, no longer in use) named www.ellopos.net/gr/grammata-ellinika.asp">GRAMMATA ELLINIKA. You can see the www.ellopos.net/gr/grammata-ellinika.asp">first chapters (pdf), but you won't find it in bookstores. I don't know of anything else besides this; keep searching, since your idea is great, and if you can't find anything, try Pharr's Homeric Greek; it uses English, but it is too a carefully written book.