"Why don't Christians obey all of the Old Testament laws?"

According to the Bible, Old Covenant laws should not be observed today because we're under a new covenant, making the first obsolete (Hebrews 8:13). Jesus fulfilled them by offering himself as a perfect sacrifice in the greater heavenly tent, interceding on our behalf (Hebrews 9:11-12, 8:25). In doing this, He abolished all dietary, sacrificial, and purity laws. (Ephesians 2:14-15). They were merely a shadow of the new covenant, temporarily imposed until the time of reformation (Hebrews 9:9-10). The New Testament is clear we still need to obey the moral law, but to revive the Old Covenant system would be a rejection of Jesus' forgiveness offered in the New Covenant (Galatians 5:4).

Why would God give such weird, harsh laws and why are they now obsolete?

The "weird, harsh" laws served two primary purposes: They helped accomplish God's evangelistic mission for Israel (Genesis 12:3, Genesis 26:4, Deuteronomy 4:5-8) and gave us a backdrop to understand Jesus and the New Covenant (Colossians 2:16-17). These laws fit into the following categories:  

Below is an explanation of why each category was given and why they're all obsolete except for Moral laws.

Why these laws were given

The sacrificial system provided a temporary solution for sin atonement (Hebrews 9:9-10). Because God is just, He must punish every sin (Romans 1:18). Yet God graciously provided the sacrificial system as a way to deal with the problem of sin so fallen mankind could have fellowship with Him. Although the blood of bulls and goats could never take away sins (Hebrews 10:1-4), Jesus validated Israel's sacrifices by shedding his own blood, "redeeming them from the sins committed under the old covenant" (Hebrews 9:15). Old Testament believers were saved by faith "on credit" until Jesus, the perfect sacrifice, came (Galatians 3:24). The sacrificial system also gave mankind a way to understand sin, atonement, and ultimately Jesus' death on the cross. Animals were constantly being slaughtered for the sins of God's people, providing a vivid picture of substitutionary atonement. It was a necessary backdrop to understand what it meant for Jesus to "offer his blood in the greater heavenly tent to secure our eternal redemption" (Hebrews 9:11-12).

 
Why we no longer observe these laws 

We no longer need priests to mediate for us or animal sacrifices because Jesus is the perfect high priest who put away sin once and for all by sacrificing himself on our behalf (Hebrews 9:26, 10:11-14, 10:18). He now mediates between God and man under a new covenant, completely changing the way we relate to God (Hebrews 9:15). When we repent and put our trust in Jesus, his sacrifice is applied to us. The wrath of God that we deserve was poured out on Jesus and we are given Jesus' sinless record. To sacrifice an animal today would be equal to saying Jesus' sacrifice wasn't enough to pay for our sins—a complete rejection of the new covenant. (Galatians 5:4). Because of Jesus' sacrifice, we no longer need laws relating to the temple, offerings, priests, or anything else in the sacrificial system.

Why these laws were given

Purity and dietary laws were given to differentiate God and His people from false gods and the surrounding nations (Leviticus 20:23-26). Many laws sound bizarre and arbitrary because they're a reaction to ancient pagan customs, diets, and worship practices. For example, it was necessary to restrict "Baldness for the dead", "cooking a baby goat in it's mother's milk" and "eating winged insects" because Israel's pagan neighbors did these things. The detailed lists of clean and unclean animals were given so God's people would eat different foods and sacrifice different animals. Had the pagan practices been different, the purity laws would also have been different to contradict them. God wanted Israel to be seen as a separate (holy) nation (Deuteronomy 14:1-21) making it clear they didn't worship the false gods of their Gentile neighbors. (Deuteronomy 14:2). This idea of separation was even reinforced with objects lessons such as "not sowing a field with two kinds of seeds" and "not wearing cloth made of two materials".

It's important to note that "unclean" was not a synonym for immoral. If it were, God wouldn't have allowed sojourners to eat unclean animals (Deuteronomy 14:21). Some unclean things were immoral (sacrificing children) but some weren't (trimming edges of beards Leviticus 19:27).  Both practices were unclean because of their association with pagan worship and customs, not because of their moral qualities. Since unclean was a morally neutral term, we rely on other parts of scripture to know what actions are immoral. We know child sacrifice is wrong because murder and worshipping false gods is clearly condemned throughout the Bible. Beard trimming on the other hand, was simply labeled "unclean" for the specific purpose of differentiating Israelites from their neighbors who trimmed the edges of their beards.

What made something unclean was it's association with either false religion or the fall of man. 

Why we no longer observe these laws

Jesus' sacrifice made Jew and Gentile one by reconciling them both to God, abolishing the ordinances that separated them (purity and dietary laws) (Ephesians 2:11-21). That's why God declared that Gentiles and all foods are now clean (Acts 10:28, 11:5-10, Mark 7:18-19); we no longer need the many purity and dietary laws intended to draw a line between Jews and Gentiles. By extension, this includes all related object lessons such as "not mixing fabrics" or "sowing two kinds of seeds in the same field". Gentiles are now "members of the same body, and partakers of the promise in Christ Jesus through the Gospel" (Ephesians 3:6). They have direct access to God through Jesus and no longer have to become part of Israel to know Him.

Why these laws were given

On a practical level, the Civil laws maintained order in Israelite society while protecting citizens and sojourners. From an evangelistic perspective, the civil laws held God's chosen people to a high moral standard so the nations would know Israel worshipped a morally perfect God. The false gods of Canaan encouraged sinful behaviors such as beastiality, child sacrifice, and temple prostitution. But the God of Israel judged his people strictly when they sinned, commanding them to purge the evil from their midst (Deuteronomy 13:5). This was necessary so the Jewish nation could be God's tool of evangelism and judgement. God's penalties in the civil law may seem harsh to us, but as rebellious sinners we have a distorted, low standard of morality. We're not in a position to tell a morally perfect creator what penalties were adequate for judging sin during His theocracy in ancient Israel.

Why we no longer observe these laws

The Old Testament civil laws were only intended for a specific people (Israel) at a specific time (during Israel's theocracy under the mosaic covenant) (Leviticus 26:46, Deuteronomy 4:1-2). It would be sinful to enforce the civil laws and punishments in our churches today since God only gave ancient Israel the authority to carry them out. For example, stoning an adulterer in your church would not only be murder, but a complete distortion of God's word. Both Old and New Testament are clear that Israel's civil laws served a specific purpose and don't apply to us now that we are under a new covenant.

Why we still observe these laws

Morality is a reflection of God's character. For example, Justice, mercy, and love are virtues because they align with who God is, while lying, stealing, and murder are contradictory to Him. Moral laws apply to all people at all times, while sacrificial, purity and civil laws only served a temporary purpose. So how do you differentiate moral laws from the others? It's usually clear from the context, but if you aren't sure you can see if it's repeated in the New Testament.