What type of server does your mail client connect to (i.e. pop3 or imap)? What pop3/imap server do you use? How do people authenticate (on the server) when they connect?
In short, yes this is possible. I use courier-imap and host a few domains. People login to the imap server using firstname.lastname@example.org
This is specified when the account is created on the server, and can be changed after creation.
If you have many accounts already on your server that you want to append the domain name to, you may want to write a script to go through all of them to do it for you.