The Mail app in Mavericks works great. My guess is you are using that non standard terribly implemented IMAP email service known as Gmail.
That may be the case, but Gmail worked great before Mavericks for me and now, Mail will simply stop checking email. While I'm not a fan of Google by any stretch of the imagination, they are probably the largest/most popular email provider in the world and Mail purports to support them natively when you set up and account. There's no denying that mail has several issues including....
1) Just not delivering mail until a restart of the application
2) Delayed loading of messages
3) Poor performance when viewing some mail attachments (probably connected with Quick Look performance issues in Mavericks)
Apple has admitted that Mail is broken and continues to try to get this right. It wouldn't surprise me if Gmail was jerking Apple around (they've done this before with Gmail).