Since Ray, in earlier chapters of How to Know God Exists, has made a case (not the best case; perhaps not even his best cause, but a case) that there is evidence for Christianity, "there's insufficient evidence" is an objection dealt with only in passing and cursorily in chapter twelve. Rather, he considers four broad categories of complaint (I've consolidated about thirteen objections he lists).
First is the problem of human (and animal) suffering if there is an all-powerful, all-knowing, perfectly good and loving God. If God knows about children with cancer, earthquakes in Haiti, tsunamis in Asia, etc. and loves us and seeks our good, and is able to do anything, why doesn't He do something about these things?
Second is the problem that the Bible presents claims about God that are hard to reconcile with the claims that He is perfectly just, that justice tempered only by perfect love. From enacting savage laws to ordering genocidal wars, God doesn't seem terribly loving or just in parts of the Bible.
Third is the problem that Christians very often do not demonstrate the holy and benevolent character that God is supposed to instill in them. Hitler claimed to be a Christian. So do a lot of churchgoers whose daily conduct, while far from genocidal or tyrannical, is also no better than that of your average non-Christian.
Fourth is the problem of whether the Bible's factual claims are more trustworthy than, say, those of the Elder Edda or Homer.
Ray deals with the first problem by converting it into another example of the second: God has cursed creation, he says, for the sins of Adam and Eve. That is, God is tormenting and killing kittens and African orphans and elderly Japanese with Alzheimer's for the offense of ancestors they've never heard of and whose actions occurred hundreds of generations before they were born. This, like declaring eternal war on the Amalekites for the wrongdoing of one particular generation of Amalekites, is hard to reconcile with the more individualistic justice extolled and promised in Deuteronomy and Ezekiel: "the soul that sins [not his son or great-to-the-hundredth grandson) shall die."
Ray deals with the second by arguing that we need to to consider the overall picture in the Bible. God, he claims, is seen as really perfectly loving when you take into account everything the Bible says about Him. Ray does not develop this idea to the extent seen in some comments to his blog: e.g. that the massacre of the Canaanites was loving and merciful because the adults were bound for Hell anyway, and this way the kids escaped Hell (one must wonder why, if this logic has such merit, Jesus was shown as so resistant to such reasoning when his disciples asked him to call down fire from heaven on an inhospitable Samaritan city). Rather, he presents a rather anemic case that, okay, God had His moments of misogyny, racism, and genocidal sociopathy, but He did a lot of nice stuff too! Ray doesn't consider, even to reject the idea, that the Bible reflects an evolving and inconsistent (across time and sometimes within a single generation) conception of God.
Ray deals with the third problem by insisting that Christians who persecuted heretics and infidels, or harassed scientists, or waged wars of conquest and oppression, were not true Christians. There are no hypocrites (or genocidal racists like Hitler) in the true Church, Ray insists, but there are plenty who look, superficially, as if they're in the church. Indeed, it would appear that true Christians are a scattered archipelago in the vast sea of false converts. One must wonder just how successful God's plan of salvation has been, or whether it bears the marks of supreme and illimitable power and wisdom, when it has won such a small fraction even of the apparent church to true repentance and faith.
And Ray deals with the fourth problem mainly through argument from assertion (e.g. the Bible was written by men who only wrote down what God told them to, so it's obviously as infallible as if God wrote it directly Himself), although he does note that the Bible is right about some things that can be checked historically and that several ancient historians seemed to accept Christian claims that Jesus really existed. And then, he quotes a passage from Josephus about Jesus that is widely considered, even by Christian scholars, to at least include emendations and embellishments by Christian copyists ("He was the Messiah," etc.).