There are several Dynamic DNS clients available for Linux. I have never had them work satisfactorily with my DNS provider ZoneEdit.
It is possible to use wget to send the current IP address to ZoneEdit (all on one line, of course). $/usr/bin/wget -O - --http-user=USER_NAME --http-passwd=PASSWORD 'http://dynamic.zoneedit.com/auth/dynamic.html?host=DOMAIN_NAME.COM'
This can be run as any user.
I have found it useful to use cron to run it. Instead of a specific time, I just use the macro @daily in my crontab.
NOTE: ZoneEdit is migrating to a user key rather than a password and, while they are maintaining backwards compatibility for now, you should migrate as soon as practical.