The "Late Antiquity" myth
A discussion (and refutation) of this phenomenon is here.
A further discussion emphasizing the purported implications for the modern United States (so often compared with the Roman Empire) is here.
I'm not going to waste space debunking the revisionist viewpoint. The linked articles above, and the book they discuss, do that job very thoroughly. But the question arises: why has such an obviously false view of history acquired such popularity? The articles argue that it has something to do with the requirements of certain modern groups' self-image. I think there may be several other, more influential factors at work.
(1) The rise of modern fundamentalist Christianity has led to a desire to "rehabilitate" the Dark Ages. Secularists have long pointed to the backwardness of that period as an example of why religious dominance over societies must be avoided. As far back as 20 years ago I was hearing arguments that the Dark Ages weren't really as bad as we think nowadays. The point then was to disparage the Enlightenment as being not such a great step forward, but by the same logic the fall of the Roman Empire has to be portrayed as not such a great step backward.
(2) On a related note, the Roman Empire was pagan for most of its history, while the Dark Ages were Christian. The identification of the former with civilization and the latter with barbarism inevitably strikes the modern Christian as somewhat jarring.
(3) Anti-Semitism has long been fueled by the view of Jews as "Christ-killers" because of their role in the death of Jesus in the New Testament. Nowadays everyone, including fundamentalists, wants to reject anti-Semitism, so there is a tendency to downplay this role and instead emphasize the culpability of the other villain of the tale, the Romans. This is then generalized to "the Romans were bad."
(4) For some reason we've developed a bias that, in any case where a smaller social group is in conflict with a big empire (usually trying to break away from its control), the smaller group is the "good guys". This bias is sustained even though in most cases it's objectively a bad thing when the empire loses. The end of the British Empire, for example, is generally considered a positive development even though it resulted in bloody chaos, oppression, and regression in many of the ex-colonies. We see the same pattern with the break-up of Yugoslavia. Even the fall of the USSR left Central Asia and the Caucasus worse off than before. Yet the general meme "empires are bad and their downfall is good" remains impervious to the evidence of what actually happens in the real world. It's inevitable that it would eventually be applied to the Roman Empire.
The fact that the fall of the Empire led to a fusion of Roman and Germanic cultures may also appeal to Americans who emphasize our country's identity as a fusion of people from many origins into one nation. In fact, this obscures a more subtle point -- the fusion of cultures usually only gives a good result when the more advanced culture is firmly in charge. The Roman Empire was already a mixture of many different peoples under a common government and a dominant language and culture. The massive influx of barbarian Germanic elements was a disaster at the time, even though the resulting fusion eventually led (in the very, very long run) to the West as it exists today. A similar Islamization of the modern West (a possible development which the article above considers analogous) would be equally disastrous, even if the resulting cultural mix produced an advanced civilization 1,000 years in the future.
The psychological needs of modern people cannot be allowed to obscure the fact that the fall of the Roman Empire was a terrible disaster for human civilization -- just as that of its modern successor, built on a continent the Romans themselves never knew, would be.