The consensus is that the idea of "Western Civilization" itself is incredibly problematic, and drawing connections between modern "western" cultures and ancient "western" cultures is possible, but you have to be really careful and do good history. That is, you need to make an effort to include the nuances and contradictions within the narrative. The textbook I'm using right now for my class, Grafton and Bell's The West: A New History, puts it succinctly: "There has never been a single, unified, easily defined place called 'the west'.....Even so, there is something we can call 'the West.' It is a web of societies, centered in Europe and its extensions, loosely linked by shared and interacting histories. These histories incorporate debts to other societies around the world and the results of conflicts both internal and global." Pretty broad, eh?
In other words, you can talk about "Western Civiliation," you just have to be careful. About what? Careful that you don't make too strong of a connection where there really isn't one. Rome didn't control all of what we now call "the West", and controlled places that we'd now mostly exclude from "the West." Islam is often excluded or even set up as a juxtaposition to "the West", yet Islamic thinkers were instrumental in carrying on Greco-Roman traditions of science and philosophy after major changes in the Empire. The Greeks, who used to be held up as the "western" standard, the founders of the humanism and democracy that defines the modern West, are being seen more as they really were: an extremely xenophobic, misognynist, slave society that had a highly restricted, if democratic, government. The bottom line is our society hardly looks anything like ancient Rome or Greece; virtually no one alive today would be super comfortable, or super safe, if somehow dropped into those far more brutal, far more segregated, ancient societies.
Perhaps the biggest problem is the way the history of the west was--and still is by some people--used for a political agenda. The original purpose of western civ classes was to create a narrative for the non-communist world after the first world war, to indoctrinate, more or less, America's young people into a proud historical tradition. This is bunk, if you take my previous point: the United States isn't really the inheritor of any Roman or Greek tradition, unless you're talking about the small (albeit important) influences of Roman Law, or representative government (we did call it the "senate" after all). Even worse, some people made it into a race thing, the idea that the ancient Greeks and Romans were "white" in the sense that modern white people are "white," and this is just ridiculously anachronistic (see my previous post from a few years ago on the history of race).
Anyway, all this is to say is that the consensus is that it's problematic. Many historians specializing in Europe will tell you that it's still incredibly important to teach the history of "the West" and will point to the various things we do inherit and the ways in which we can teach it properly; many historians of non-western history point to our global/international society and (rightly, I think) express the need to teach history surveys beyond Europe. In this day and age, many departments only get to force students to take one history class, and more and more are siding with World History. That's more reflective on the consensus than any argument among European historians.
Edit: Source: Check out Gilbert Allardyce, "The Rise and Fall of the Western Civilization Course," AHR 1982. It's old but has a decent history of the course.
