It is easy to see patterns in noisy data, but there is no evidence that such patterns make useful predictions
Business cycles exist and that seems to be a fundamental property of complex systems like the world economy (as argued by Paul Ormerod in Butterfly Economics). But the cycles are not regular enough to be useful for predictions as this review says:
In many ways, the term “business cycle” is misleading. “Cycle” seems to imply that there is some regularity in the timing and duration of upswings and downswings in economic activity. Most economists, however, do not think there is.
And this is the fundamental problem with most of the supposed evidence about K-waves: the analyses are all about fitting historic data to the supposed pattern not about making reliable and bold predictions about the future. We have had plenty of time for those predictions to be made and for the predictions to be tested and yet the believers are still adjusting the theory to make it fit history rather than making useful predictions about the future. Spectral analysis as reported in http://www.escholarship.org/uc/item/9jv108xp (mentioned in another answer) isn't much use for predicting and will tend to find more patterns than exist in noisy data. Even that analysis doesn't think k-waves explain more than 5% of the variation, which isn't much help in a forecast especially when you also take into account the vagueness of the actual cycle length (which visually looks anything between perhaps 45 and 60 years).
Commenting on some of the original predictions by Kondratiev himself one critic notes:
Kondratieff postulated a "long wave" of business that began somewhere in the late 1780s – it is all very murky since there are almost no statistical data for that period – and continues periodically roughly every 54 years. Well, what about the trough points? No question that the late 1930s – a "Kondratieff trough" – was a pretty miserable period. But what about the other three trough periods? What was wrong about the 1780s, for example? No particular depression there. And if we want to be generous and dismiss that "first trough" for lack of data or as only starting the whole thing, what about the alleged second trough? Fifty-four years from 1789 brings us to the "expected" trough year of 1843, a year in which everything was smooth sailing. Let us be generous and bend over backward for the Kondratieffites, and give them their admitted 1849 as the trough year. Even so, 1849 was a perfectly fine economic year, and in no sense whatever comparable to the late 1930s! In 1849, we were in the middle of continuing prosperity.
The third alleged Kondratieff horror point, or trough year, was 1896. But, again, there was nothing terribly wrong with that year either. Of all the trough years, or even trough zones, the only one that we can really say was bad and depressed was the late 1930s: One out of four!
and later talking about more recent predictions (he wrote in 1984):
But the Kondratieffites' problems have only begun. Their real difficulties come after the alleged Kondratieff trough of 1940 – the last trough so far. The entire boom-bust "long" cycle is approximately 54 years in length. Allow a few years here and there. But still: It has already been 44 years since the Kondratieff trough. A 44-year boom! So where's the peak? The peak is getting long overdue. Most of the Kondratieffites confidently predicted that the peak would arrive in 1974, just 54 years after the previous peak. Previous peak-to-peak stretches had been 52 (from 1814 to 1866), and 54 (1866 to 1920). So where indeed is the peak? It is now 1984 and counting. We are ten years past the confident prediction and we still have inflation.
There is a sound reason in human psychology why we keep believing the predictions of these theories: we remember predictions that seem to work and forget the other predictions of the same experts that don't work. So an expert can attract a body of evidence of successful prediction by making a number of alternative forecasts and then, after the event, erase the forecasts that were wrong. This problem is extensively discussed by dan Gardner in his recent book Future Babble, also by William Sherden in his book The Fortune Sellers and by Nicholas Taleb in Fooled By Randomness (all of which should be compulsory reading for all skeptics). Their key message is that it is really easy to be fooled by apparent patterns in noisy historic data, but they are rarely if ever true and even more rarely valuable when trying to make decisions about the future.
In short Kondratiev cycles predict nothing useful.