The origin of this particular falsehood is a passage in the Bible (1 Kings 7:23): "He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it."
The accompanying story with this is (usually) that the government somewhere, often in a southern state (usually Alabama) has made it the law that pi must be three, because (of course) all Southerners are so stupid that they believe the Bible is completely true, word for word.
In Indiana in 1897, a law was killed in the State Senate redefining pi similarly. It is possible that this is the origin of this rumor.
The viciousness of this rumor is something I don't want to comment on, but it's interesting to note that nowhere in the passage does it say that the shape was an exact circle or that it was exactly thirty cubits in circumference (a very loose measure as is).
You should also take a look at Pi in the Bible.