Having done quite a bit of "mechanic-ing" over the years, it's certainly true that newly assembled engines are usually quite tight, so are certainly best treated carefully until they begin to "bed in". Many of the moving parts need to do this and as pointed out by Davy B above it isn't just the engine. He's quite correct that brakes don't achieve their maximum efficiency until bedded in either which is again a good incentive for careful treatment from new. Don't forget that the gearbox is perhaps the most complex mechanical part of the car and this also will benefit from careful treatment whilst it's tight. However this isn't to say that the car needs to be driven slowly for this period, normal road speeds won't do any harm, which is where the 'myth' that they don't need to be "run-in" comes from, although I certainly wouldn't "thrash it" as suggested above!
My advice would be never to work the engine hard from a cold start - this applies whether new or old anyway as the oil doesn't circulate as efficiently when cold. Keep the revs low until the engine has warmed through thoroughly. Engines aren't harmed by high revs, but don't use full power and load the engine heavily whilst it's new. Above all, and by far the most important, don't 'labour' the engine by loading it at low revs in too high a gear as this will do far more damage than revving it a bit higher and keeping the loading on it light. I'm a firm believer that if you look after it, it is more likely to continue to run reliably than if you place much wear on it's components by pushing it too hard when it's very tight and especially when it's cold.
I worked at Honda for a few years and the technicians always said run them in hard, it won't do any harm and would loosen everything up ,you get better fuel consumption and the ECU learns the way you drive it.
After about 15 brand new cars and doing this I've never had a problem so I guess it's each to there own. I must admit though I never have a car after its warranty runs out
Planning on running mine in for the first 500 miles anyway.This is my first 'brand new' car. So that means gentle acceleration and braking, not turning too harshly, and not taking the engine over 3000RPM (IIRC that's about 90kph/55mph in 5th).