There is (or at least was) a place to set up a new mail notification sound within Thunderbird itself, and I think you need esd for it to work. However, this also means that you must set Thunderbird to check for mail periodically and also leave it open all of the time.
My computer is too old and slow for that to be an option, so I'm using another program to announce new mail (I've also switched to Slypheed-dev, it works better on my system). There are quite a few mail notifier programs that you can set to start when fluxbox starts and then run in the fluxbox slit. For example, there are a several listed at dockapps.org
I tried a couple of different things out before settling on GKrellM, which has a lot of different, useful functions, including a mail notifier that will play the sound of your choice when mail arrives. In the GKrellM setup GUI, you need to specify a command to play the sound, like aplay your.file
or play your.file
("aplay" is part of ALSA and "play," which I use, is part of the sox package).
GKrellM is included in Slackware. You can find updated versions at LinuxPackages
also, along with themes and additional monitors.
So, yes, there are several ways to get a new mail sound with fluxbox and any email client, including Thunderbird.