First off, this quesiton is rigged because the question is exclusive.
Anyway, I would still say that God/religion is the only way to ground morality. Think about it, what are the alternatives?
If you exclude the possibility of a God, you need to provide a reasonable explanaiton of the universe. The most common explanation is evolution. I am not saying that evolution is false, and this is not an argument that I am making against it. If you believe that evolution is the ture cause of the human race, then you must accpet the fact that we evolved slowly. If you believe we evolved slowly, then you must also accpet that our brains evolved. If you beleive our brains evolved, then you must beleive that our sense of reason and morality also evolved, for where else did we get intuitive senses of right and wrong?
This also begs the question: is morality innate, or is it taught? If it is innate, then it should be the same for every person and should be consistent. In the concept of evolution, Morality isn't objective, so it only follows that it is not inate.
If it is not innate, then it must be taught. If it is taught, then it is subjectve. If it is subjective, then it only follows that it is not objective. If it is not objective, then there is no firm basing for Morality according to evolution. However, not everything that is taught is subjective, for instance, physics isn't subjective. Physics is taught, so not all things that are taught are subjective.
However, we must look at wether or not Morality can be tuaght. Can morality be defined? No. It cannot. Much like virtue, it cannot be defined. If it cannot be defiened in it's truest form, then it cannot be truley taught. If it cannot be truley taught, then It only follows that it must be inate and not subjective.
However, evolution shows it to be subjective. Therefore, you must have a diety, or higher, inate sense of what morality is.
Basically morality is the only thing that allows society to function (THIS IS NOT REFUTABLE), however there are several different theories as to why it exists be it evolution or as you put it a transcendent concept created by god. All of them apply some form of logic to prove themselves so this question is null and void.
There is only one Truth, but there are many ways to perceive it
Historically, religion has been the strongest source or motivator for morality. But modern ethics is determined not to base itself or be dependent on religion. Imagine Socrates and Dostoyevsky debating this; Dostoyevsky was a Russian Orthodox Christian and Socrates was an agnostic. Dostoyevsky is known by his statement, "If God does not exist, everything is permissible". To him, God and morality is a package deal, if God falls, so does morality. In other words, if God does not exist, there is no objective goodness. Socrates would argue that religion is not the only ground or justification for ethics, morality can be grounded in a rational philosophy. Now imagine that Euthyphro joins the discussion; Euthyphro ascribes to the divine command theory. For him, morality is grounded in the will of the gods. Dostoyevsky is similar to Euthyphro in the sense that he says yes to morality because he says yes to religion, but instead of divine will as the source of ethics, Dostoyevsky says that it is God's nature or essence that is the source of morality. I personally agree more with Socrates, I think morality has many logical grounds or justifications other than religion. Religion may be the psychological source of one's ethics.