Logging in with a non-gmail account
You can only log in to the market with either your gmail account, or an account hosted with Google Apps for your domain. 

This is not just to authenticate you, information is synced between your device and google servers. If you don't have an account with them, they don't know who you are. You said that you can login with your personal email address (someone at mydomain.org). I am going to guess that mydomain.org is hosting the email on google apps for your domain.