I think this really, really depends on make/model.
On the newer vehicles I've dealt with, I haven't had any problems that could be considered specific to newer vehicles. My household's Honda, Mazda, and Subaru (all from within the last 5 years) have/had easily accessible service points (fluid fill holes, oil dipsticks, drain plugs, filters) and are fairly easy to get around. My '07 Civic was about as easy a vehicle to get up on 4 jack stands as you could hope for. All three vehicles also have relatively affordable access to the official factory repair manual (either in book form or online) to boot.
Lots of stuff is computer controlled, which could become a problem down the line. But as previous posters have said much of that represents DIY work that is no longer needed (e.g. plug wires, ignition timing, etc.). I guess my verdict is that on the new vehicles we've had, anything that falls under typical DIY maintenance has been easy enough, although it's possible that some repairs down the line will require professional intervention for at least part of the job.
I get the impression that some more expensive cars and luxury makes are a different story though. I'm not a fan of engines that don't come with dipsticks, especially from companies like BMW.