No.

With bolt action hunting rifles of similar quality I don't think that you will find a lick of difference in accuracy between a 30-06, .308, .243 7mm etc.
I think with a high end custom made target rifle, the .308 might gain a bit of an edge however if only because of a slightly stiffer receiver but more so from less recoil and probably from a shorter column of powder to burn as compared to the .30-06.

I think the reason .308's got a reputation for greater accuracy was because of comptetitive shooters and the rifles they used at the time the .308 was introduced. the >308 was introduced in the early '50's when rifle competition was dominated by M1 garands. It was found that rebarrelling and rechambering the M1s to .308 gave immediate and significant gains in accuracy. The main reason for this was the rifle itself and not the cartridge. The M1 has an oprod that is very sensitive to port pressures, so 30-06 loads had to be kept in a narrow range which meant that the cases were not loaded anywhere near to capacity or with the best powders fo rthe cartridge, the .308 on the other hand, favors powders that work well in the rifle and has a much better load density than the 30-06. I belive thi shas caused the .308 to get a reputation as a more accurate cartridge.

AS an example of how the rifle effects the cartridge, most people who have shot it, including myself consider the 6.5x55 to be one of the most accurate cartridges out there. This cartridge is chambered in swedish military rifles that are of very high quality and fit that they put modern production hunting rifles to shame. I've heard of people buying american made rifles in the cartridge being somewhat underwhelmed by the accuracy as compared to the cartridge's reputation, but then, American manufacturer's don't guage, carefully inspect, and hand lap every barrel they screw into a receiver.