Do Clear Alcoholic Drinks Give You Less of a Hangover?

It appears to be true.  Research as far back as 1960 suggests that drinking whiskey is more likely to give you a worse hangover than if you were to drink vodka instead (1).  A more recent study replicated this finding (2), and both seem to suggest that it is the presence of congeners, impurities that result from the fermenting or aging process, that are to blame.  The latter study, which came out just last year, tested sleep quality, performance on cognitive tasks (the day after drinking), and self reported measures for hangover symptoms (headache, upset stomach, etc.).  What they found was that people who drank bourbon reported suffering from more severe hangovers than people who drank vodka (with both groups ingesting quantities that gave them equivalent scores on a breathalyzer).  While drinking either type of alcohol resulted in poorer sleep and poorer cognitive performance, there did not appear to be a difference in these measures when comparing the bourbon drinkers to the vodka drinkers.
Of course, to truly know if it is the congeners that are to blame for the hangover severity, rather than some other difference between vodka and bourbon (like the former is made from potatoes while the latter is made from corn), perhaps another study comparing silver to gold tequila, or "grain" ethanol (distilled to be 95-100% pure)  with varying levels of acetone, aldehydes, or other congeners added in.
However, if you want to know which drinks contain the most congeners, the generally accepted list goes something like this, Brandy and Red wine contain the highest levels of congeners, followed by brown and dark Rums, and Bourbon and other whiskeys (including Scotch).  White wine, clear Rum and Gin have pretty low levels of congeners, but Vodka has the lowest levels of any spirit (except for "grain").  That being said, you should avoid the cheap stuff and go for the "top shelf" selections, which are likely to have been distilled 3 or 4 times and thus contain less impurities.