I think I need to clarify on the Old Testament laws, and their application today. This is by no means a complete look at OT laws, it's just a slice of some of them that helps us find perspective. It will mainly focus on laws that we look at today and think "That's bad". There are a lot of other ones we may say "That'd odd" or "That's pointless" to today, but I won't be covering them here.
Matthew 19:1-12 contains a conversation between Pharisees and Jesus. It basically goes like this:
Pharisees: "So, like, is divorce ok for any/every reason?"
Jesus: "No"
Pharisees: "Then why does the Law of Moses say it's ok?" (see: Deut 24:1-4)
Jesus: "That Law was given because of your hard hearts, but it was not this way from the beginning."
Yes, I missed a lot, but this is the part that pertains to my point. Read the passage, the rest doesn't change the scope. God has a view of marriage - it is not in His design for separation to occur in any circumstance*. However, God knew that the people of the time would not accept that law, so He gave them a more relaxed law, that he knew they could obey. Namely, divorce only if something "indecent" is found about her, and no remarrying if she's remarried and divorced/widowed.
It's very clear that the laws given in Deut regarding this are less strict than that given by Jesus. "Indecent" changed to "Marital Unfaithfulness". Has God's standard for marriage changed? No, but He believes our hearts are softer regarding this - so He gives us a more strict moral standard, one which holds up to even today's standards*. Indeed, this is much more moral than today's standards, which state, "get a divorce whenever you want, really. It's cool." (If you disagree here about what is more moral, it's probably because of different views of what love and marriage are and entail, so don't jump on it straight away.)
This can't be the only Law that was given that catered for the people of the time, certainly not. Take the slavery laws - Deut 21 vs Ephesians 9. For the times these laws were given, slavery was pretty much the global civic. It happened. The sermon on the mount (Matt 5) is almost exclusively dedicated to redefining the laws in a more moral sense - ones that stand up to even our greater global morals today.
So the most that can be said about the apparent harshness of the laws in the OT is that it was given in a context where people had hard hearts, and wouldn't accept laws that were stricter than those.
Even with the Jews' hard hearts, the laws given were better than what else was around then. You cannot equate laws of 5000 years ago with laws of today without taking cultural maturity into consideration. The laws were good, for their time. Now, well that's another kettle of fish.
Furthermore, the 10 commandments are in play. The laws given aren't the focus, the commandments are. The first 2 commandments alone are all that is necessary for all the other commandments, all the Old Testament and New Testament laws. Think of the 10 Commandments as the "What" and the Law as the "How". Others see the 10 Commandments as "God's" law, and Moses' law as the "State" law. Church / State separation perhaps?
When it comes to today, the laws still have to be interpreted via the 10 commandments, and must pass the "filter" of New Testament teachings - I say filter, because nothing is taken away*, only narrowed down and simplified. Remember, God's Law hasn't changed, but our ability to respond to them has.
While I say nothing is taken away, a lot of stuff is rendered irrelevant. "Masters, love your slaves" obliterates the relevance of "If your slave loses an eye because of you, they can go free". Of course, if you're respecting them, then they won't lose an eye in the first place. "Don't Murder" is supplanted by "Don't be angry with anyone". And both of these are covered by "Love your neighbours as you love yourselves", and more so, "Love your enemies".
Stop focussing on OT law. It's much clearer in the New Testament, from the same source (God), and is more catered to the state of our hearts. The very laws that govern our society come from these roots, and there's very-near parity between them. Saying God's laws are immoral, while touting our society's laws is a lesson in contradiction.
If someone has objections about the morality of the standards put forth by God to us today as explained in Bible, I would seriously question their motives. The standards put to us (and lived out by Jesus to show as possible) are far better than any other set of moral laws ever invented.
Footnotes
1. Yes, it says except for marital unfaithfulness, so I'll grant that exception. But then, I don't think unfaithfulness is in God's design for marriage either, so it's a moot point.
2. Though, admittedly, there is controversy over this too - what about an abusive partner, for example? That's another argument which won't be covered here, as it's a bit off-topic.
3. Not strictly true - the ritualistic cleanliness / worship ones are taken away, for example, because that context for worship is gone.