It's a little harder to prove that myths about ancient history are false than it is to disprove myths about more modern eras, but the prevailing opinion is that the following ideas are wrong.
The following ideas about ancient history might more properly be called "urban legends" to signify that they are mostly modern ideas about ancient history.
In addition to the following list, there are plenty of myths the ancients wove into their history. To learn about these, start with Introduction to Greek Mythology.
Hundred Greatest Men, The. New York: D. Appleton & Company, 1885.
The idea that Julius Caesar was born by Caesarean Section is old, but since Caesar's mother, Aurelia, was involved in his upbringing, and the surgical techniques of the 1st (or 2nd) century B.C. should have left her dead, it is unlikely that the story about Caesar's birth by C-section is true.
Akhenaten was an Egyptian pharaoh who put aside the traditional Egyptian pantheon of gods in favor of his own sun god, Aten. He did not deny the existence of other gods, as a monotheist would have, but held his god above the others, as a henotheist.
The date of Akhenaten may make it impossible for the Hebrews to have borrowed from him, since their monotheism could have preceded Akhenaten's birth or followed the return of traditional Egyptian religion.
We don't even know for sure what year Jesus was born, but references in the Gospels suggest Jesus was born in the spring. The god Mithras was born of a virgin on the winter solstice and was known as the light of the world. It is likely that aspects of the worship of Mithras were adopted by early Christians.
4. Latin is the Most Logical Language and Superior to Others
This is a hard one for me since I tend to buy into this myth, but Latin is not any more logical than any other language. However, our grammar rules were based on the grammar of Latin. Since English is, but should not be put into a Latinate mold, English comes out looking awkward. The specialized vocabularies we use in areas like law, medicine, and logic, tend to be Latin-based, too, which makes Latin seem superior.
"Beware the leader who bangs the drums of war in order to whip the citizenry into a patriotic fervor, for patriotism is indeed a double-edged sword."
The quote is anachronistic in detail and spirit. There were no drums and all swords were double-edged. The idea that citizenry needed to be persuaded of the value of war is not from the first century B.C.
Atlantis was used as a parable by Plato and possibly mentioned by Solon of Athens. Whether there might possibly have been a real lost continent of Atlantis or not remains open to debate, mostly among non-academics.
It is believed that when the person in charge of a gladiatorial event wanted one of the gladiators to be finished off, he turned his thumb down and that when he wanted the gladiator to live, he pointed his thumb up. The editor's gesture signifying that a gladiator should be killed is not exactly thumbs down, but thumbs turned. This motion is thought to represent the movement of a sword.
The Amazons were probably not the one-breasted man-haters we think of when we hear the word. They are more likely to have been fully-breasted Scythian horse-riding warriors, judging from artwork, although Strabo does write that their right breasts were seared off in infancy.
Aside from the question of whether the U.S. is designed to be a democracy instead of a republic, there are countless differences between what we call democracy and what the Greeks did; furthermore, it is totally unfair to say "all Greeks voted" or to claim that those Greeks who didn't vote were branded "idiots".
At the Battle of Thermopylae there were 300 Spartans who lay down their lives to give the rest of the Greeks a chance, but there were a total of about 4000 fighting under Leonidas, including willing Thesbians and unwilling Theban allies. Read more about the Battle of Thermopylae.