No, the email notifications should work perfectly fine with gmail. I would recommend that you make sure that you have your desired alerts set properly at http://www.chess.com/home/alerts.html
Also, you should make sure that you allow "@chess.com" as an approved sender to your email service so that emails from Chess.com are not filtered into the spam folder.
And also please note that while you are logged into the Chess.com website you should be receiving notification boxes on the actual website rather than email notifications so that it cuts down the emails that Chess.com must send out.
For some reason I get email notifications very sparatically. Sometimes never, sometimes right away. I've played 5 or 6 games and it's always been this way. I have a gmail account for email. Is this the reason? Do you guys not work well with gmail? Thanks