You are totally right, after enabling SMTP auth, it works like a charm, and postfix accepts it.
Now I have to find out how I can have the Exchange 2007 machine in our office accept mail from that email addres.. It now says "501 5.1.7 Invalid address (in reply to MAIL FROM command)", because the from-address is "".
Can I set that sender address somewhere in the CMC? Or any smart ideas for now on how to have Exchange accept that? I will google.. If I can't find anything, I can always try to do some magic with the postfix in between (like send to a local account on the postfix machine, and forward it to an account on the exchange).