It seems to me that the modern outlook has three essential components, which began to appear centuries before the Enlightenment:
1) Nothing has a basic character or essence that ought to be respected (nominalism)
2) Nothing can be known except by direct sense experience (empiricism)
3) A good society is one in which everyone's desires are satisfied equally (utilitarianism)
All of these premises have a certain inevitability, once you start thinking in the right direction. If the reality of universals is denied (there are no "chairs" and "triangles," just individual objects that we arbitrarily describe as "chair-like" or "triangular"), then there is no prevenient metaphysical order for our minds to apprehend. Since the universe is made up of unconnected particulars that can't be understood as part of a coherent whole, then we have no reason to take a stance on any metaphysical statements (like the existence of God) and had better stick to what we can confirm by our senses. And if there is no meaningful metaphysical order to the universe, then it's hard to see how we could justify any theory of "virtue" or "justice" - we're left with the ideal of maximized individual pleasure as the sole standard for ethical behaviour.
Besides its logical soundness, these three statements all appeal to people's direct interests. A government policy aimed to maximize the fulfillment of everyone's desires will be popular almost by definition, and a politician foolish enough to campaign on the opposite platform ("I will create policies that prevent people from fulfilling their desires!") would never reach any public office. Even #1, which seems like a bit of abstract metaphysics, is really about human freedom and self-determination. If "humans" have a set of invariable universal characteristics, then the basic model for human flourishing becomes an ecological one: we have a particular niche in the world, a particular set of goals to which we are oriented and in the pursuit of which we become most truly free. Such a model means that our ability to determine our own destiny is secondary: we are defined by our responsibilities to each other, to the rest of the world, and to the Supreme Being. If, on the other hand, there is no such thing as "humanity" but only individual humans, then we have an infinite scope to determine our own destiny and priorities.
The problems with this view of the universe are legion, and from this point the critique of modernity practically writes itself. A society built along such principles is ultimately rootless and nihilistic; when humans have infinite scope for freedom and no sense of their own limitations, they will inevitably abuse their power, at the expense of their natural environment and anyone who gets in their way. By applying an abstract calculus of "equality" to flesh-and-blood people, modernity inevitably dissolves the ties of family, national identity, and religious belief that hold us together. And because the desires of individuals and groups inevitably conflict with others (my right to set off an air-raid siren in my backyard conflicts with your right to practice the recorder next door), liberal societies have difficulty mediating between the conflicting interests of different groups in a successful way.
If you agree with this criticism, however, if you think there's something wrong with rational utilitarianism as the basis for a civil society, it's not immediately apparent what you should do about it. If you were to suddenly abandon all traces of the language of Enlightenment rationalism and spend your days loudly repudiating the entire liberal intellectual tradition since Duns Scotus, you would have no point of contact with the general culture and would soon have very few friends. Such an extremist stance insists on the total depravity of modern culture, and is thus really a form of secularized Calvinism; it ignores the real good that has been done by humans in the past five hundred years. Yes, the Third Reich would have been unimaginable in a pre-Enlightenment world, but so would Die Zauberflote. The question, then, is how we can repudiate the tendencies that gave us Hitler while retaining a connection to the culture that gave us Mozart.
This is a hard question, probably the hardest that our society currently faces. We're very far from an answer, and I suspect that at this point in history it may be impossible to find one. A real intellectual sea change, like the original Enlightenment, would likely surprise everyone and seem inevitable only in retrospect. In the meantime, antimodern ideas still have almost no place in the public square, and political debates, to the extent that they involve actual ideas as opposed to name-calling, are essentially turf wars between different forms of liberalism.
Consider, for example, the recent American health care debate. The justification for creating a nationalized health care system, from the point of view of modernity, is that access to universal health care allows individuals to pursue their own interests in a free and equal manner. This logic was opposed by the Republican party, on the grounds that the increased regulation and taxes inherent in such a system prevent individuals from pursuing their own interests in a free and equal manner. From my point of view, there is nothing to choose between the two positions, both of which assume the gratification of individuals to be the highest goal in life, and the rearrangement of society to facilitate this goal the end of political activity. No-one (that I'm aware of) made the point I wanted to see made - that it seems highly questionable whether my strep throat or your broken leg should cost hundreds of thousands of dollars and require the coordinated involvement of either corporate or government bureaucracy.
Even if the "conservative" parties were to take a wholly anti-modern and reactionary position, it's not clear that this would do any good. Suppose that our Reactionary Party was somehow able to get elected and set to work undoing all legislation that reflects Enlightenment individualism (supposing the unlikely case that they were able to agree on what "reflecting Enlightenment individualism" entails, and that there were no judicial obstacles to their project). Such a project would have no long-term success, since the basic outlook of modernity would persist, and would give rise to the same liberal political ideas as soon as the Reactionary Party lost power. The Reactionary Party's policies would also be fundamentally incoherent, as rolling back recent legislation would simply return us to an older form of liberalism - the nineteenth-century free-market individualism described so heartwarmingly in Oliver Twist - which no-one is particularly keen to see return.
Similar problems face reactionaries outside the political realm. Composer George Rochberg is celebrated in First Things for starting a "revolution" (an interesting word choice) for his abandonment of integral serialism and a return to the tonal language of the late nineteenth century - using late Beethoven, Mahler, and early Schoenberg as jumping-off points. Yet the language of chromaticism, which arguably led us into the modernist cul-de-sac in the first place, seems unlikely to lead us back out. Rochberg's late works are certainly beautiful, and they likely served a useful purpose in encouraging a more humane attitude toward composition in the academy - but we're looking for an escape from modernity, not a retreat to its more comfortable enclaves.
Our old problem - what to do as a traditionalist in an anti-traditional world - is neatly summed up by George Grant:
Life teaches one all kinds of ironies. I think irony is necessary if you're going to have your car fixed, and you can hardly live without a car in in twentieth-century North America. How can one live as a Christian in a modern university without irony? If a Christian spoke frankly in a modern university, he would have to leave. Yet is one just to give up these institutions? Therefore I don't think irony is dishonest or wrong at such points. All of you know that.English-Speaking Justice, xix.
This is about right, I think; neither a head-on assault on modernity nor a withdrawal from civilization, but an ironic engagement with society that allows one to undermine modern ideas in a more subtle manner. In the end, after all, there is less riding on the makeup of civilization than we might think - whether our society endorses them or not, the old ideals of truth, beauty and goodness are still there. They've never gone anywhere.