It may seem old-fashioned, but I think this is the best strategy for learning such a highly inflected and essentially dead language, especially if you only have experience with mostly analytical languages to begin with. Much of what makes this work is the work of genius that is LLPSI, but I have come to see the value of this strategy for language learning in general. If you are as convinced of the input-hypothesis as me, then you might agree that this is the quickest way to get reading independently in Latin, which is where most learning will take place.
The only objection I have with Dowling's method, is that you should start reading LLPSI immediately, rereading chapters over and over while you are memorizing the tables. Only pace yourself, and don't rush on to chapters covering grammar that you haven't started memorizing yet. If you get bored with old chapters, pick up the companion books, like Fabellae Latinae, for more reading materials.