Can't send email from a computer that is not on the same domain as mail server

We have just set up a mail server on Amazon. I can send email from the command line fine. I can send and receive email from a PC connected to the Amazon domain with Thunderbird. I can receive email from a PC outside of the mail server (Amazon) domain with Thunderbird, but cannot send mail. I always get a timeout error from a PC outside of the Amazon domain. The SMTP settings are the same between the computers connected to the different domains. All of the necessary ports are also open. Any help would be greatly appreciated.