The catch with all of this is that for many days after the change, some email will trickle to the old server until EVERYBODY'S DNS gets updated. It could take 5 days.
You need to find a way to query the old server occasionally over the following few days to catch the straggling mail. At this end, I could query spunky.mail.dreamhost.com for my mail, or its IP address.
To figure this out, find out the IP address of your current mail server using the 'host' command from a UNIX shell. Then use that IP address in your mail program.
OR set up mail forwarding at your old place to forward mail to off-site accounts such as Gmail or the like. I like this idea better. Once your old site shuts down for good, all mail will end up at your new host.