I don't know if its a question of "should." People certainly can talk about morals without invoking God. The conventional wisdom, though, is that God is required to live the moral life and develop a moral society.
But there's a Slovenian philosopher called Slavoj Zizek who quotes (Hegel I think but I'm not sure). But Zizek, quoting this other philosopher, says: the typical formula is "If there is no God everything is permissible." But Zizek says that's actually wrong: "If there IS a God then EVERYTHING is permitted; but if there is NO God then NOTHING is permitted.
To my way of thinking, this latter conception is the correct one. There is a sense in which the use of God as a crutch (I'm not saying everyone who believes in God is using It/Him/Her as a crutch) actually trivializes life, because no matter how you screw up, as long as you, say, "accept Jesus (for example, I know you didn't mention Christianity specifically) as your Lord and Savior" in time, then you can go to Heaven. There's a sports analogy I like to make here, but this answer has already gone on a little long; so I'll leave it there.