Modern materials technology makes it less important, but not unimportant.

On the older vehicles, it's still vital.

I have had several new vehicles in my life and the only one that came with such instructions was my Harley. The others just came with a bit of verbal from the dealership.

I confess, GavinBKK, that in spite of a complete lack of break-in instructions, I was (because I always am) fairly gentle for the first few and subsequent miles. Decent gas mileage requires gentle anticipatory driving. Given gentle driving, I usually get long life from my tires and brakes as well. I don't mean that I'm one of those slow pokes that drive you nuts at traffic stops, I just don't floor it for a getaway or slam on the brakes to stop it.

I used to have a toy car which had a sticker in the back window saying "Running in", which my father said cars used to do in the 50s and 60s.
I have seen only about four cars in my life with such a sticker -- one of them recently on a Rover P6, and unlikely to be still running in.
http://en.wikipedia.org/wiki/Rover_P6

Modern engineering tolerances are such that the need for running in on new "hardware" is non-existent. (Whether that applies to Detroit.... ;-) )

Three decades back or so, new Detroit cars came with a "break in oil" in the engine. Drivers were advised to drive easy and to not maintain a constant speed (vary speeds) for the first 1,000 miles. After that, you would change out the break in oil for your regular oil and install a new filter. Besides breaking in the engine, this was supposed to make sure that any medal shavings or other debris left over from the manufacturing process were removed.

My last couple new ones, going back to 1990 did not come with those instructions.

Also gone are the instructions to let your car warm up at idle for 5 minutes before driving.

Of course, AEH, the reason for the last is pollution-driven. A cold engine idling doesn't heat up the catalytic converter fast enough to meet pollution standards.