One way that is still used to denigrate and mock Christianity, as well as the ancients and medievals, is the suggestion that, prior to Columbus, everyone thought the Earth was flat. This belief was rooted in religious dogma and was therefore unchallengeable until it was demonstrated empirically to be false; and even then many people continued to affirm it. It is held up as a primary example of the folly of religion in contrast to the wisdom of science.
I fortunately grew up knowing that this story line was bogus. People did not think that the Earth was flat before Columbus. Every educated person from about the third century BC onward knew the Earth was round. Columbus was trying to discover an alternate passage to the East Indies by sailing west. He had to convince people that such a route would be superior to the common one of going south, around Africa, and then east; but he didn't have to convince anyone that the Earth is round. Besides, how exactly did Columbus's voyage prove the sphericity of the Earth? He didn't circumnavigate the globe; he didn't reach some place traveling west that had already been reached by traveling east. Isn't it obvious that this narrative is false?
I thought that these things were fairly well-known. I suspected that anyone who seriously thought otherwise essentially got their knowledge on the subject from Bugs Bunny cartoons.
(Update, 30 March 2012: Here's another proof via Bugs Bunny that the earth is round.)
It just amazes me that people take this urban legend seriously. I think, for example, of the globus cruciger, that ball with a cross on top of it that kings would hold. The ball was supposed to represent the earth, with the cross on top representing Christ's dominion over it, and the sovereign would hold it to show that "he's got the whole world in his hands." The earliest of these dates to the fifth century, before the fall of Rome, and they were used throughout the Middle Ages. In fact, orbs without the cross were common for centuries beforehand. Thus, any claim that the ancients or medievals thought the earth was flat can't even get started. You can see plenty of pictures of them online, and you can watch a short documentary on the globus cruciger here.
Unfortunately, there are still people, including historians (so I can't lay the blame on the side of popular culture), who believe that Columbus was trying to prove the Earth is round. The go-to book to refute such claims is Jeffrey Burton Russell's Inventing the Flat Earth: Columbus and Modern Historians. There are also some excellent resources online: see here, here, here, here, and here, for example. James has pointed to a recent book promulgating this claim which may indicate a new trend: using the "flat earth myth" to impugn Christianity and make Islam look better by comparison.
Regarding the Bible, there are passages which refer to "the ends of the earth" and "the four corners of the earth." However, they do not amount to an assertion that the earth is flat anymore than our use of terms such as "sunset" and "sunrise" amount to assertions that the sun revolves around the earth. "The ends of the earth" merely refers to the most distant places, and "the four corners of the earth" refers to the most distant places in the four directions in which one can go (north, south, east, and west).
Regarding Christian history, there are a few historical figures who went against the flow, but this does not negate the consensus view. The extent to which a flat earth was accepted in ancient and medieval Christianity is sometimes exaggerated based on criticisms of the theory of "antipodes." But this seems to be a misunderstanding: "antipodes" referred to people who were alleged to live on the other side of the earth. The Christian authors who rejected this (not all did) pointed to the almost universally-held belief that it was impossible to travel from one side to the other, "either because the sea was too wide to sail across or because the equatorial zones were too hot to sail through" (Russell). Therefore, no one from one side of the earth could have gotten to the other side, so that if there were people on the other side of the earth they could not share a common origin with us. Some have unfortunately taken these statements to mean that they were denying there was an "other side" of the world at all. But these authors were making anthropological statements, not geographical ones.
The only individuals who clearly affirmed a flat earth were Lactantius (third and fourth centuries), whose "views eventually led to his works being condemned as heretical after his death" (Russell); Severian (fourth century); and Cosmas Indicopleustes (sixth century) who exerted virtually zero influence on his contemporaries or the Middle Ages: "The first translation of Cosmas into Latin, his very first introduction into western Europe, was not until 1706. He had absolutely no influence on medieval western thought" (Russell). By way of contrast, Copernicus translated some short writings of Theophylactus Simocatta from Greek to Latin in 1509. While this was the first such translation published in Poland, and thus had some importance in that regard, the text he chose was not. The reason he chose Theophylactus is because all the good stuff had already been translated, so he had to settle for the dregs. Cosmas wasn't translated for another two centuries. To suggest he was even taken seriously by the handful of people who read him is just absurd.
Additionally, Diodore of Tarsus (fourth century) and Theodore of Mopsuestia (fourth and fifth centuries) are referenced by other Christians as affirming a flat earth in order to refute them, but the actual writings in question are lost. Isidore of Seville (sixth and seventh centuries) is often given as an example of a flat-earther, because some of his writings seem to affirm corollaries of a flat earth. But since he also gives a figure for the earth's circumference (80,000 stadia) and affirms that the sky is spherical and equidistant from the earth on all sides, it is difficult to attribute a belief in a flat earth to him.
So Lactantius, Severian, Cosmas Indicopleustes, Diodore, and Theodore of Mopsuestia make a grand total of five Christian writers who affirmed, or apparently affirmed, a flat earth, all of whom lived in late Antiquity at the very latest, and none of whom were taken seriously.
So how did such a silly idea become so popular? According to Russell, it goes back to about 1830 when Washington Irving published his story of Columbus, and took some license with the historical account. In Irving's story, Columbus wasn't trying to discover an alternate route to the East Indies by sailing west around the world: he was trying to prove more basically that the Earth is round in the first place. Before this time, everyone thought the Earth was flat because that's what the Bible teaches. Columbus's detractors were the priests and inquisitors who didn't want anyone challenging their authority to proclaim what reality was or wasn't.
Despite the absurdity of these claims, by about 1870, western society had pretty much uncritically accepted the idea that everyone thought the world was flat prior to Columbus's voyages (including, ironically, some Christians who took it upon themselves to defend flat-earthism). There were two primary reasons for this naïve acceptance that the ancients and medievals thought the earth was flat: First, the 19th century was a time of great optimism for the human race. People thought that we were quickly advancing towards a manmade utopia, and for many this implied the superiority of modern man over his predecessors. Thus, it was very conducive to this worldview to portray those who lived prior to the Enlightenment as a bunch of uneducated half-wits who didn’t even know the earth is round. World War I pretty much eradicated the optimism, but much of the disrespect for and contempt of our predecessors remained and remains still.
Second, at this time, some people were very confident that scientific discoveries would eventually explain everything without any recourse to God (naturalism). However, many scientists did not accept naturalism, so a cultural campaign was initiated which sought to identify it with science itself, and to this end represented any denial of naturalism as part and parcel of ignorant religious believers getting in the way of truth and progress. Examples were found, twisted, and sometimes completely invented in order to illustrate the point. The flat earth was a perfect candidate for one of these "examples": in Irving's story, he had made Columbus's opponents the priests and inquisitors who didn’t want anyone challenging their authority to make pronouncements about what constituted reality. Indeed, a lot of naturalism's credibility comes from the degree of absurdity in examples of what religious people believe or have believed about the physical world. When this degree of absurdity turns out to be misinformed -- either totally invented or significantly misrepresented -- naturalism no longer appears as obvious.
So the flat earth myth isn't just an urban legend; it's propaganda, deliberate misinformation that is presented in order to prop up a position without going through the tedium of finding actual evidence for it. It doesn't bode well for your worldview if you have to change reality in order to make it fit.
(cross-posted at Quodlibeta)