I admit I have a hard time wrapping my head around people who say they want to play a cleric with no ties to the religious part. I mean, it's in the title. The word "cleric" has a clear definition: a priest or religious leader. That's sort of the core feature of the class, like magic users using spell books. If you want to play a class that has similar abilities but not the religious part, I get that, but it should be a different class altogether. Kind of like the sorcerer is to the wizard.
Saying you want to play a cleric without the religious part is like saying you want to play a Colonel without dealing with any military stuff. It's counter to what the word literally means.
Religion does not necessarily mean a particular deity, however. For instance, Buddhists usually don't believe in gods, but you'd have a hard time claiming Buddhism isn't a religion.
That said, non-theistic religions shouldn't be a way to get around having some sort of behavioral restrictions on clerics. Again, using Buddhism as an example, the requirements of that religion on its disciples are at least as stringent as those of Christianity or Islam. The difference there is that they are phrased more as "This is how to lead a good life" than "Do what I say, or else."