Thanks for the replies, think I have this issue resolved now.
Yes, with the default emaill client, there is a small icon (looks like an envelope) that stays there after the initial notification. Guess I had not really noticed it at first. And yes, by dragging down, I get a list of the unread mails.
My final solution, at least for now, was a free alternate email app called Emoze. I like the look of it better than the default app, and it comes with a nice big widget that can be placed on the home screen to show unread mail. Maybe the widget is a little too big (4x2), but I am satisfied. The default app is also not bad, just a matter of how you like your email to look.