I'll more or less echo joeqwerty. However, you'll want to set up a temporary, secondary MX with the new IP address before the switchover, so you can avoid delays in email delivery. Find out what the TTL (cache lifetime) for your DNS is set at, and make sure that the time difference between adding the MX and changing the IP is greater than that. Or, just turn down the TTL to an hour or less, to prevent clients from caching the DNS for too long -- again, do this well before the changeover.
So in addition to updating DNS, you'll need to change the external IP address of your router, or of the server if it is directly connected to the Internet.