Ars Technica reports that Apple is still trying to figure out a way for certain apps to do it. There's two options laid out here: one is user-based, in which the user actually approves certain apps to run in the background, with the consequence (if you do choose to enable that) of lower battery lifetime. The second has Apple approving apps for background action, allowing certain developers to run based on limits of "resource usage such as RAM or network bandwidth."
Obviously, there are pros and cons to each plan -- giving the power to the user means they will be able to choose when the battery is drained (on a particularly busy day, for instance, or when the iPhone is plugged in), but it also means that users will have the ability to crash their own phones (allowing too many background processes could cause issues). And of course, while leaving the choice in Apple's hands will make sure background processing is only used in the "right circumstances," we all know how great Apple is at app approval.
Ars concludes on the same point that I would: if background processes are really going to make a difference, they'll likely rely on a future iteration of the iPhone, as the one we've got now is just not built to run apps all the time, no matter who chooses them. Even if Apple is working out a way to run certain apps in the background, they are almost certainly spending more time beefing up the iPhone hardware as well.