One of the built in apps that will run in the background is the Microsoft Push Notification Service (MPNS). Windows Phone Developer Blog has two blog posts, part 1 and part 2, that go into details on how the service works.
Microsoft's solution revolves around Windows Azure. Let's say you have a Twitter app and you want to be notified each time someone mentions your @twittername. As I understand it, your Twitter app registers with the Azure server and lets it know it wants to subscribe to mentions. MPNS gets alerted from Azure and the phone either shows notification toast - a popup, or the app tile on the home screen updates its display with new information.
For the most part, this is what multitasking on a device is all about, but without the penalties of having battery life or system performance issues. It doesn't solve all issues of course, and Apple has addressed these shortcomings in 4.0. Maybe Windows Phone 8 will do the same, but I digress.
My concern is the stability and reliability of the Azure servers, or any cloud service. By running everything through one cloud service like this, there is a risk that there will be downtime that will affect virtually everything. When Gmail goes down, you cannot get Gmail, but Skype still works. However, when Blackberry servers go down, since virtually everything on the BB runs through it, the devices largely revert to simple phones. If (when?) Azure goes down, all push notifications from third party apps on a WP7 device will cease to function until the service is back up.
Stability isn't the only issue. Actually, I expect the servers to be pretty reliable even if we don't see 100% uptime. The other issue is, once a developer stops supporting an app it may well stop functioning if it drops out of the Marketplace. Apps may essentially die on you, which is a relativly new experience for computer users.
Overall though, I am not totally sold on a lack of multitasking. We'll have to wait and see how the execution works when WP7 launches.