I have exactly the same issue. Even when I have my WiFi stopped, looks like when the phone is sleeping the 3G radio goes off and I stop receiving e-mails.
Basically in order to receive an e-mail, I have to specifically turn the screen on and manually check my e-mail.
That is really a big issue I see here. On every other phone I had I would receive my e-mails when they come. Here, not only that the stock app does not support push e-mails, but also now the 3G radio goes off and basically receiving e-mails via mobile data becomes a strictly manual process?
Is there any why that can be fixed. Maybe installing an app or something.
All I want is when I have my phone in my pocked, if I receive an e-mail, to get notified. And not needing to check my e-mail manually every 10 mins?
Thanks very much
Tsvyatko