Arab Light crude is about $25 shy of its 2008 all-time high (both in nominal and real terms), and it has been rising steadily since last September. As this chart shows, every recession since 1970 has been preceded by a sharp spike in real oil prices. Does the recent spike mean the economy is headed for another recession? I think there are good reasons why oil prices will not necessarily precipitate a recession.
To begin with, the aggravating factor that precipitated every recession on this chart was a significant tightening of monetary policy, as characterized by a pronounced rise in real interest rates and a very flat or inverted yield curve, as illustrated in the above chart. Today the monetary situation is completely different; real yields are very low and even negative, and the yield curve is quite steeply sloped. Rising energy prices can be problematic when the Fed is tight, because tight money puts the economy on a strict monetary budget: when you pay more for energy you have to pay less for other things. Thus expensive energy can really squeeze the economy when money is tight, and it is the shortage of money that results in a recession. Today the Fed is quite accommodative, which means that the Fed is essentially working hard to ensure that expensive energy prices don't result in a shortage of money to spend on other things.
The other reason is that our economy has become much less dependent on energy over the years, thanks to technological advances and conservation initiatives. Energy expenditures represented about 9% of all personal consumption expenditures in 1981, but today energy only consumes about 6%. Even though energy prices today are higher in real terms than they were in 1981, we are spending about one-third less of our incomes on energy today than we did back then.
Update: To clarify my position on higher oil prices: they are a drag on growth because they make economic activity more expensive/less profitable on the margin, but at the same time higher oil prices put more money into the pockets of oil producers, who must then spend it on something else. Higher oil prices do not cause money to go down a black hole; dollars spent on oil produced by OPEC countries never leave the U.S., they simply change hands. Higher oil prices will slow growth, but at these levels they don't threaten a recession.