It's probably not excessively difficult to play an aetheist game. The key would be controlling Happiness through resources such as Wine (Monarchy) and Dyes (Calendar), and making Drama(Theaters) and Construction (Colosseums) a semi-priority. The latter two buildings will provide a great deal of Happiness depending on where you set your Cultural slider (probably far more Happiness than you'll get out of religions, even at as low as 20%). Additionally, Theaters provide additional +1 Happiness to Dyes.
As for losing Priests, I don't see that to be much of a problem. You need either a heavy investment in infrastructure (namely, you need multiple religions per city for multiple Temples per city in order to run multiple Priests, or you need Shrines, or you need Angkor Wat, or any combination of the above). The hammers/commerce they provide can be easily duplicated elsewhere (probably to greater effect, with the exception of Priests under Angkor Wat, The Sistine Chapel and the Representation civic).
You may want to build The Pyramids for earlier Hereditary Rule (+1 Happiness per troop per city) and Representation (+2 Happiness per top five cities). Seeing as you'll be looking at an early Hereditary Rule anyway (Monarchy, aka Wineries), you could even skip The Pyramids and still do fine. Additionally, you can switch to Nationhood (+2 Happiness per city with a Barracks) later on.
I think the hardest parts of such a game are going to be making sure to not discover a religion (crazy, huh, that you'd have to actively attempt to not discover one), and putting up with the eventual (constant, I'd think) requests to convert to a religion.