sending mail to a server alias from a local email account doesn't work

[Dave Martini]

I configured sendmail on RHEL 5.4.
I can send mail when I'm logged into the RHEL machine to
root at hostname.llnl.gov
and I recieve the mail in my Thunderbird inbox since I have
root defined in /etc/aliases to go to martini1 at llnl.gov.

But when I try to send a message from my Thunderbird inbox
email system on a different machine to root at hostname.llnl.gov
I never receive it.

Is there some other configuration I need to do with sendmail
in order for this to work? I want to be able to send mail
from any reader on any machone to aliases that are defined on this RHEL 
5.4 machine.

Thanks.
Dave Martini
LLNL