One of Mail's ongoing problems has been the lack of the "New Mail" sound being played when new messages arrive. There have been numerous fixes that have surfaced for tackling this problem, and while they will work for some people, the root of the silent New Mail sound problem has not yet been pinpointed so the fixes do not necessarily help everyone.
My recommended approach for this problem is to create an alternative to using the built-in notification for new mail by setting up a rule to play sounds when messages arrive. Other options are to use a notifier like Growl (though this software has its share of quirks), and to use third-party junk filters that support sound notifications. Recently MacFixIt reader and hbmug member (yes, MUGs are still alive and kicking) "Ken" wrote in with his solution to use SpamSieve as yet another solution to the missing New Mail sounds.
Install SpamSieve (it is a $30 purchase, but may be worth it).
Copy the "New Mail.aiff" sound to the /Macintosh HD/Library/Sounds/ or /username/Library/Sounds/ folders.
Use the SpamSieve preferences to select the new mail sound (or any other sound you wish).
Alternatively, he suggests that if the Mail sound file itself has some corruption or odd attributes applied to it, then it could be what is preventing it from playing when new mail arrives. Though this is a possibility, I do see it as a remote one; however, it will not hurt to replace it as a suggested fix.
Install Pacifist and open it (it's shareware, so you will have to wait 15 seconds if you do not have a license).
Insert your Snow Leopard DVD and open it with Pacifist.
Locate the "Contents of Mail.pkg" in the OS X installation disk, and go to the Applications/Mail.app/Contents/Resources/ folder.
Select the "New Mail.aiff" file and either extract it to the desktop and then move it to its default location in the Mail application folder, or use pacifist's "Install" option to do this automatically.