http://bugs.winehq.org/show_bug.cgi?id=31438
--- Comment #12 from rkfg eurekafag@eureka7.ru 2012-09-03 07:30:11 CDT --- Oh, I've added that without comments. Sorry, here's the explanation.
If you're just want to play ME3, you actually can do it by downloading the installer, unpacking the game and importing some data into registry. The link to the ZIP can be acquired from the console output when launching with WINEDEBUG=+winhttp, it's hosted on akamai so seek for "akamai" there or grep it. The link should include the auth key which is unique for your account or session. Shutdown Origin, grab this link with wget and then unpack the archive to "/Program Files/Origin Games/Mass Effect 3" in your Origin prefix. Then import the data above into registry with regedit /S
Now start Origin and you should be all set. The game will complain that it should be updated, just click Not now; if you choose to update it will indifenitely prepare download or check because of the bug. I suspect it may be related to files and not the network since repairing the install leads to the same indefinite preparing.
You can download and unpack any DLC the same way, just restart Origin and let it check the files, you'll get 100% instantly and DLC should work.