To make an obvious statement, people use different media differently. That is, the frequency in which someone checks email is different than the frequency in which they read their Twitter stream ( I certainly know people who check Twitter more than they check email, and I've had DM or Facebook message conversations that have replaced email - but that's not really the point). I think generally users need to be able to decide which media alerts are interrupt driven, and which ones are meant for background occasional browsing.
For certain applications that are vying for your attention, I think engaging your users via email to spur a return to your application on an clear opt-out basis has become totally acceptable, and for me anyway, is generally preferred. Twitter and Facebook have largely paved the way for making this the de-facto method, by alerting you via email when there are new followers or friends, or Direct Messages, and allowing you to opt-out if you don't want them. I probably wouldn't know about any of this activity if it weren't delivered by email. I certainly wouldn't want this info delivered as a tweet itself for the Twitter examples, anyway.
|Twitter notification by email|
I don't think it was always this way. It used to be considered a bit spammy to make an application email it's users without consent, but I think it's become somewhat accepted, again, as long as there is a clear opt-out of "I don't ever want to get this email again" right in the first email.
Many applications have taken the same tack with iPhone push notifications, and that's something I don't like. I really don't want my phone beeping every time someone writes on my Facebook wall, but if I remember correctly, this is the default behavior.
The general point here is, think carefully about how to drive continued engagement with your users, and although with any notification medium you will always walk a thin line between engagement and being spammy, don't be afraid of email. It's still the primary way most people communicate.