First, remove any file that contains private data. That means files in
/srv, anything in
/etc that you've customized, logs in
/var/log, mail in
/var/mail, typically many other things under
/var, etc. Only keep some minimum network configuration, in particular SSH keys if needed (you may want to create throwaway SSH keys for the duration of the wipe, and remove the others).
Overwrite and unmount all partitions other than the root, e.g.
# you shouldn't have any open file on non-root partitions at this point
</dev/zero cat >/dev/sda2
Now fill in the empty space:
</dev/zero cat >/zero
At that point, most of your data is gone. With some filesystems, there may be some data left over in the last block of each file (e.g. on a filesystem with 1kB blocks, if a file is 2000 bytes long, then there are 48 unused bytes at the end); on ext4, I think this unused space is always zeroed.
For a final this-is-goodbye pass, terminate as many services as you can, then overwrite the root partition, e.g.
</dev/zero cat >/dev/sda1
Your system won't boot back.
Don't bother with overwriting with multiple random passes instead of zeroes: this advice is based on 1990s disk technologies and no longer relevant; today, zeroing is just as effective, and even then, recovering zeroed data required expensive equipment and quite a bit of luck. See Overwriting a Disk and follow the links for more details.