In short, dispensationalism is a theological system whereby God is said to have different laws and expectations for different people at different times. There is, for example, the Dispensation of the Law when the Old Testament Law was in effect. There is a dispensation pre-Law, when obviously people where not under the Law because, you know, there was no Law yet. In this system the Law is ended or suspended from the time of Jesus until some future time; the dispensationalist theologians argue over precisely when this will be. What this does is excuse Christians from having to follow the Laws of the Old Testament because they were not meant to be applied to anyone after the coming of Christ. We hit the cosmic reset button and get a whole new set of rules. Actually we just get two: Love God and Love Your Neighbor. Allegedly. Stay tuned, first century A.D. Christians, for more rules and regulations regarding sex, speaking in church and other fun stuff...just no killing...unless you're a heretic! What dispensationalism does not address is the acts that we would consider immoral today are still being commanded by, or in some instances carried out by, God.
One of the rationales that many people use to justify immoral acts by, or in the name of, God, is to suggest that the people who the Israelites committed genocide upon were worshipping false gods and therefore deserved their fate. Of course, if today someone was murdered because they worshipped the wrong god, or worshipped in the wrong way, we would be full of righteous indignation and condemnation. Compare Old Testament behavior to what Islamic terrorists do today to see what I mean.
And lest we think that it was just those evil unbelievers who were on the short end of the biblical stick, there are plenty of rules that call for horrific behavior against "God's People" as well. It was legal to sell your daughter into slavery. For that matter, slavery was legal and encouraged. Beating of slaves was regulated. Stoning of children for disobedience was legal. Executing people for minor offenses was typical.
The most common rationale is to just assume that whatever God did is by definition moral because God did it and then ignore the specifics. It has been my observation that most people have little knowledge of the Old Testament other than some of the stories (like Noah's Ark, The Exodus and David & Goliath) and three or four of the Ten Commandments and therefore don't think too hard about what it contains. The exceptions are some more conservative folks who selectively use it to justify positions that go against the cultural flow, such as racism 50-60 years ago and opposition to equal protection under the law to gays today.
It is easy to see that the morality of the Old Testament is an understandable outgrowth of the time and culture, and many bible-believers understand this. What they don't understand is that the New Testament, the messiah-ship of Jesus, Christianity itself, and its conception of God, is built upon the foundation of the Old Testament. If they deny the God-derived origin of the Old Testament, they are by extension, denying the God-derived origin of the New as well. They have hewn the legs out from under the argument that their faith, and therefore any morality contained therein, is "truth". It's either handed down by God or constructed by Man. If one wants to hang onto the argument that it's handed down by God, then the evidence that the morality that most people adhere to is superior to that outlined in the bible must be accepted and all that entails. If one wants to relegate the morality of the bible to culture and custom, then any claim to having "the truth" must be abandoned, and the believers are in the same boat as the rest of us: defining our faith by what makes sense, what "works"for us and what stories we choose to tell ourselves.