I found this really simple way of finding the installed BIOS version on an Ubuntu PC without having to reboot. Simply execute the following command in a terminal session and scroll through the output until you find the BIOS section.
sudo dmidecode -s bios-version
For more system information, just scroll through the output until you find what you need.
sudo dmidecode | more
Update August 2012
I have successfully installed Lubuntu 10.04 on an old Toshiba Tecra 8000 (Pentium Mobile 233 with 256MB of RAM) and found that this trick to find the BIOS version did not work. The BIOS in my old Tecra is older than 1999 and doesn’t have the Desktop Management Interface present.