Your @me.com emails should be accessible at http://icloud.com (if necesssary click the cloud icon at top left, and choose 'Mail' from the window which opens).
If you have set up iCloud correctly on your Mac you should be able to access your emails in the Mail application.
Thanks for the answer. Unfortunately, my iCloud account was not setup correctly because iCloud could not decide which of my two accounts was the correct one. Thus, for some time mobile.me allowed me to send mails using the old account that did not allow to receive a message.
The problem is solved for me. But others may still have problems.