The first thing to note, is the importance of having an axiom. For the non-theistic case, I'll use the axiom mentioned above: 'suffering is bad'. However, there is also an axiom (however well-concealed) in the theistic case, of 'God is good'.  That axiom is rather misleading however, or at least complicates things unnecessarily, as we can get on to a lengthy debate of ontological arguments, and fundamental properties. Instead, let's change it to 'God is truthful', which ultimately fills the same function: those who believe God is the source of morality either believe God said it, or wrote it in human nature. In either case, if we can't take God at his word, how do we conclude that morality holds? Now, to examine this axiom. 

Do we have any reason to assume God is truthful? Not really. We could use the ontological argument to show (arguably) that God is the pinnacle of goodness: even accepting that conclusion however, how do we then conclude that being truthful is an aspect of goodness? If God is the source of our moral knowledge, any answer is suspect. If God is dishonest, then any statement of his that truthfulness is morally good obviously cannot be trusted.  This is discussed partially here, though this post will go into more detail. Similarly, even if we can show unambiguously that truth is the morally good option for humans, how do we so much as touch what is morally right or wrong for a deity? Gods are an entirely different class of creature, and one no one pretends to fully comprehend: how does it follow that, even if honesty is morally good, that it is also the right thing for a deity to do? 

This lack of any reasonable comparison between human and God also answers the typical response: why assume a human tells the truth to you?  With humans, we can assume some element of human nature, to say nothing of simple pragmatism (no one wants to be caught out on a lie). None of this is the case with God.  And for more religion-specific arguments, God commanded Abraham to kill Isaac. That would seem to be a moral statement: that, in this case, killing your son is a good thing. It wasn't the case, however: the faith required might have been good (debatable), but the action apparently was not, as God prevented it. At the very least, God here is misleading.  In short, we cannot take for granted that God is truthful. 

Can we assume it? Of course; which makes 'God is truthful' an axiom in the purest sense. Something taken as fact, because it is a necessary first step, and not because it's backed up by anything. 

All of this, is simply a long-winded way of putting the theistic and non-theistic axioms of morality on equal footing.  Neither one is better than the other. Any moral statement that appeals to 'suffering is bad', compared with one appealing to 'God is truthful', while the latter may seem more impressive, on fundamental terms they are of equal weight.  This is an unfortunate fact. Morality is, unavoidably, subjective to some degree. Objectivity is unattainable. 

For an illustration of this, we'll look at Euthyphro's Dilemma. This is an (admittedly discredited) argument against divine morality, and goes: "Is something good because God commands it, or does God command it, because it is good?" In the former case, we have what is, on the surface of it, meaningless morality: on the second we have God who is as much a subject to morality as anything. There are responses to this, but it is better to phrase the dilemma more generally, appealing to a non-specific standard of morality A, which could be God, or could not be: "Is something good because A contains it, or does A contain it because it is good?" In the first case, we have reduced A to an axiom. In the second case, we have some other standard of morality, say B, which may or may not be tied to A (for example, the typical answer to Euthyphro is that God's nature is the ultimate arbiter of morality). Regardless, we get: "Is something good because B contains it, or does B contain it because it is good?" Using the same example, is something good because it's in line with God's nature, or is God's nature in line with it, because it's good? And now we have a cycle that can only be terminated by declaring a standard morality, somewhere down the line, as an axiom. Either that, or we appeal to another standard C, and D, and E... Which needs to end sometime, because a moral infinite regress will end up a circular argument. 

To sum up what's been said so far, all morality (divine or mundane) is, unavoidably, based on an axiom.  Technically, this is all that needs to be said. Theistic morality is no better than non-theistic morality, it merely does a better job at dressing up the flaws: we can, however, go a step further.  Can we show non-theistic morality is better, in any sense? 

Believe it or not, there are a few ways to do this. The quickest is to look at the matter of validation.  It's easy to find a theist arguing that, for example, the Bible's (misnamed) Ten Commandments give incredible moral advice, that everyone will agree with. They then proceed to list 'Thou shalt not kill' and 'Thou shalt not steal' as objective, unquestionable evidence.  Not once have I heard anyone use 'Thou shalt not work on the Sabbath' or 'Thou shalt not take God's name in vain' as a similar example. This is especially noteworthy because many theists hold that God gave all humans a conscience, or some form of internal morality fundamental to each of us: if this isn't explicitly held, there is usually some element of it believed in. If, instead, they believed everyone was inherently evil, why would they give 'thou shalt not kill' as evidence of moral knowledge when, if we were all free of morality, many of those listening would disagree?  In short, we have innate morality. This is a scientifically explained, and otherwise undeniable, fact. (the post linked to is imperfect, but explains enough). Some ignore this 'conscience', true: the many drives that make up a human will occasionally overwhelm each other. This is far more understandable in a non-theistic world where morality stems from unguided evolution, and less so in a theistic world where it's wrought by a divine, perfect hand.  Regardless, theists appeal to innate morality as evidence of God: they never bring up many rules which are equally important to God, but have no such inbuilt agreement.  Why would God make people unwilling to steal, but willing to work on the Sabbath?  We're in the position where morality universally agreed upon, is also the morality supported by non-believers. It makes the theistic arguments that the Bible, say, contains divine morals come across more as seeking validation, by grouping 'thou shalt not kill' (a non-theistic moral statement also) along with 'thou shalt not take God's name in vain'. 

This also ties into my next point: the vast majority of religions out there share a central kernel of moral rules, and then add more and more unique to that religion, built around the shared kernel.  Even going right back to ancient cultures who practiced human sacrifice, such atrocities were justified by making it in the name of something good: making the Sun rise, for example.  Every religion is centered around a handful of specific moral rules, many of which would seem to be justified by the axiom 'suffering is bad'. This might be evidence of a deity working behind the scenes, as many Christians (for example) hold, if not for the fact Christianity is built identically.  Around all the thou-shalt-not-kills, there are Sabbath prohibitions and bans on graven images: extras added to the central moral kernel, making it indistinguishable (on this level) from the above mentioned religions. 

Back to moral relativism, for a moment, the primary criticism leveled at it is that one person's views are as valid as another's, when it comes to morality. In theory, this might be true: in practice, it is not. Practical applications are always very different to theoretical.  In practice, not only do we have to deal with simple pragmatism: we have to deal with overwhelming agreement.  'Suffering is bad' is as close as we can come to a universal moral statement. Even those that support the suffering of some, almost invariably, are doing it to lessen the suffering of others (even if that other is themselves). A great deal of intellectualizing and thought needs to be put into the axiom to construct a full-fledged system, but that's beyond the scope of the post: suffice to say, it's possible.  And regardless, as shown with the axioms above, all morality must unavoidably, ultimately, fall into this trap. 

What makes 'God is truthful' a better axiom than 'suffering is bad'?  I don't expect an answer to this question: it cannot be answered. Any knowledge we have about God, which is required to make some form of response, ultimately would (in the theistic worldview) come from God: and so cannot be trusted to give a good picture of the deity, by the trait under question.