On a vacation a few months ago I turned off automatic email fetching on my phone to try and eke out a few more hours of battery, and just kind of left it there to see what would happen.
On an average day I hit the "get my email" button twice. Once when I wake up in the morning to see if there's anything pressing, and once on my way home from work as I'm walking towards the train.
I have absolutely not missed the disruption that having a constantly-up-to-date email client in your pocket creates.
I have also yet to miss a single email due to this - people have better sense than that. If they need a quick response they know to call me or to text.
From this I've started pruning the apps and services that can interrupt me pretty hard. At this point I'm down to my calendar, my SMS client, FB direct messages, and the NYTimes (they seem to be pretty responsible, but they are on a short leash). It's worked out. I'm very much enjoying this pull lifestyle, instead of push.
why not do it then? it's pretty simple to do?
I think the real reason is that you like the idea of doing that, but not the reality.
Cultural atavism might sound like a good idea, but the reality is the positive benefits of email and smartphones way outweighs the negatives.