Actually, it occurs to me that we don't really even have to have the most generic hypothesis about what a God could be like. Most religions make FAR more concrete claims about what their God is or does. And quite a bunch of that stuff is not only falsifiable, but testable too.
In fact, I'd even say that to be of any use to a theist group or worth giving a damn about it if you're not one of them, a religion HAS to be a bit more concrete than just acknowledging that some form or another of vague and undefined creator of the universe may have existed.
A God that's of any use needs not only intelligence (which was mentioned already), but also intent, and a propensity to act in some way or another according to that intent. (You won't get many people joining in your new cult of Apatheia, the Goddess who won't help you, smite you, or even judge you, because She truly can't be bothered to care, much less do anything.) You want it to have some rules or preferences at least, and you want it to be at least somewhat predictable in applying them. (Not much point in trying to appease a God who's purely random in whether he'll smite you for adultery or reward you for it, or for any other action, thought or state.) Etc.
True, there are some, like for example the Deists, who basically acknowledge such a deity, but when was the last time one of those told you that his non-intervening outside-his-own-universe deity will smite you for allowing gays in the army? See what I mean? You can pretty much ignore that kind of gods.
So, anyway, the ones we have to worry in practice are the Gods which, we're told by their fanboys, send earthquakes and tsunamis, answer prayers, judge all the time like Dredd, perform miracles, and invariably bestow great magic powers upon charismatic con-men.
And thanks to statistics and sociology data and medical data, etc, some of that stuff is more testable than many people realize. We're not longer in an age where we could only look at some individual peasant's prayers remaining unanswered, and we can imagine that maybe only that was a fluke, or maybe only that peasant didn't qualify. We have statistics. We can actually test whether the most religious communities recover more often from just about any illness or other problems than us heathens, for example.
That just moved it from a philosophy problem, to a problem that actually can be run through the scientific method.