Some common moves:
- Leibniz: as Jpers36 has said, one can assume that God exists and is omnibenevolent, omniscient, and omnipotent--from this one can deduce that this is the best of all possible worlds (given a few very plausible assumptions).
- Deny omnipotence: I seem to hear often from contemporary Christians that God works through physical things which are imperfect tools. Despite being rubbished in the condemnations of 1277, this is highly plausible to many folks who consider this problem today.
- Redefine good/evil: this is Jpers36's solution. In its standard presentation, good and evil are poorly defined, and it is assumed that any rational person exposed to the argument will simply agree that suffering is evil, and some of it does not have among its causal descendants any greater good (or, at least, that the preponderance of evidence suggests that this is the case, even if it isn't certain).
These moves lose a lot of the flavor and meaning of God, but do defuse the problem (though it may cause other problems for any particular individual's belief set, as it may be inconsistent with other beliefs s/he holds).
The third move ends with a tug-o-war of plausibilities, and I think most people find it quite unsatisfying. Jpers36 recommends natural order as the greater good served by the existence of suffering. S/he further notes that it is not obvious that suffering actually constitutes an evil needing to be overcome. My impression is that it is much more plausible to most readers that suffering is inherently evil than that natural order is inherently good, and so this tack is persuasive to few readers. At the very least, we might expect Jpers36, or anyone who chooses this line of argument, to give us some account of exactly why it is that the proposed greater good is in fact a good, or the sufferings of the world aren't really evil.
My own perspective is that, to the extent that I have any reason to believe in the existence of good and evil at all, I have reason to believe that suffering is evil; furthermore, it seems absolutely too much to swallow that all of the myriad, seemingly pointless sufferings that occur in this world are actually each necessary for the fulfillment of some greater good. There are an extremely large number of stories of apparently purposeless woe--to give just one example, a friend of mine's father, who has always seemed to me to be a virtuous man, sleepwalked onto and then off the edge of a roof several years ago. The fall mostly paralyzed him from the waist down. If one doesn't already have reason to think that a God exists, there is absolutely no evidence to support the belief that this was appropriate or productive of some greater good--it looks for all the world like undeserved suffering.
Even if the preservation of natural law is inherently good, the argument must go further to claim that this is the best of all possible worlds which have as much natural order as this one has. On the surface, that seems relatively implausible. Either the universe is deterministic, or it isn't. If it is deterministic, then one must abandon all notions of free will and personal responsibility as they are usually conceived. Further, it still seems as though there could easily be deterministic universes which would be better than this one. If the universe isn't deterministic, then it seems as though God could intervene at any point at which determinism fails, and select the better choice. Even if He never involved Himself in the free choices of mortals, quantum mechanics seems to suggest that He will have an extraordinarily large number of options--for Him to be unable to do better than He has done seems badly implausible.
One other problem with natural law as an inherent good--miracles. Since Jpers36 noted that the bearer of value in the scheme proposed is not the particular instances of natural law working and their consequences, but the preservation of natural order as a whole (textual support for this: "The problem does not lie in any other possible evils caused farther down the causal chain; the problem is in the very disruption of the chain,"), any miracle at all would destroy the value of having natural order in our world. Since unnecessary suffering is only allowed because of the goodness of preserving the natural order, it must be the case that the natural order hasn't already been broken--that is, no miracles have ever occurred; God has never directly intervened in the world.
An altered version of this objection is available, in case the Jpersian type wishes to abandon the total sanctity of natural law, claiming only that the more often it's obeyed, the better. In this case, it would still be true that any single violation of natural law deprives the universe of some fixed amount of goodness, which means that the cost, in terms of goodness, of any miracle is the same--call it X. In order for any miracle to be worthwhile, it must bring about greater goodness than X. That's all fine, but it also means that any time a miracle is withheld, the amount of evil which could be prevented by a mircale is of a lesser magnitude than X. In other words, every miracle that has ever happened caused a greater gain in the balance of good over evil than the loss caused by each preventable evil. So if one truly believes that there have been miraculous sightings of the Virgin Mary, each viewing caused more good than a miracle preventing the worst earthquake that ever happened would have prevented. So far as I can tell, this makes it very difficult to believe testimony of miracles that aren't extraordinarily important.