It's as if they've forgotten that Jesus Himself said (in so many words) to follow the law of the land (when He said to pay taxes to Rome), and followed the law of the land even up to His death on the cross.
Erm... Yeah, that isn't actually what that particular episode was about, or a good representation of Christ's own views on political authority.
Christ lived at a time when there was no real separation between Church and State -- that is a much more modern view. At that time, the Roman Emperor was seen as a sort of deity, with his own personal cult and exalted status (despite the antics of emperors like Nero and Caligula, almost certainly exaggerated by their enemies). In some cases, they were even seen as miracle workers; Vespasian was said to have miraculously healed the sick (sound familiar?). Similarly, the King of Judea was traditionally a position with both political and religious significance, including the support of the Temple (which, at that time, was the central religious authority for the Jews.) Jesus grew up in an occupied land, lorded over by Roman outsiders who were causing all sorts of socioeconomic disruptions in Judea. Rome did allow Jews to maintain their religion, but treated the Jews as second-class citizens even in their own land. Roman governors were actually cruel and dismissive of the populace; the depiction of Pilate in the biographical gospels is, to put it mildly, at odds with every scrap of historical information we have about him and his fellow governors. Worse yet, the Romans co-opted the Temple and Jewish leaders. Many Jews thought that the Temple had become corrupt, resulting in competing movements such as the Essenes and Pharisees.
Rebellions were also common in ancient Judea around that time -- and often violently suppressed by the Romans. In addition to the two major revolts after Jesus' life, there were several messianic uprisings around 4 BCE, including in Sepphoris. Less than a decade later, the Zealots attacked the Romans at Sepphoris, and after they were defeated, the Romans crucified 2000 of them. Sepphoris was only a few miles from Nazareth, so there is no question Jesus heard about these events.
Christ's own mission was radical -- he basically wanted to toss the bums out of the Temple, rewrite the Covenant, and bring the Kingdom of God back to the world. While this does not
seem radical to us today (as we don't live under suppressive Roman rule), it did then.
And what of rendering unto Caesar? If you read the passage, it's pretty clear what is happening. The Pharisees know that Jesus does not recognize Roman rule, and that an open proclamation of a rejection of Rome will get Jesus arrested and executed. They ask Jesus whether it is right to pay the Imperial tax. Jesus reacts angrily, as he knows it's a trap. He demands to see the coin in which the tax should be paid, which is the Roman (i.e. outsider/foreign) denarius, with a depiction of the Emperor on it. Jesus is basically saying "this coin is worthless, so Caesar can have it back. The ultimate authority is God, not Caesar." However, he said it in a way that did not openly insult the Emperor, thus he escaped the trap.
Keep in mind that in that era, coins were not just a means of exchange; they were often a form of propaganda and a political identifier. Coins often announced changes in the Imperium or lauded critical victories. One of the first things the Jews did during the First Revolt was to print their own coins, where the year of the revolt was "Year 1."
For hundreds of years after the execution of Jesus, Christians were in revolt against Roman rule and suppression. It was only much later (in the time of Augustine iirc) when the debate changed, as more Romans adopted Christianity. Theologians and church leaders wanted the official status, support and legitimacy denied to them for so long. To do that, they had to drop the adversarial and prosecuted position, and accept Roman rule, and become a part of it. As a result, the Church reinterpreted not just passages like "Render unto Caesar," it changed or lost the earlier oppositional context of Jesus' ministry.
In modern times, almost all of that political context is lost. This is due to a general lack of historical awareness, as well as concerted efforts to remove all of that political context from Jesus. How can he make statements intended to last thousands of years, when he was merely reacting to local political events...?