Alright, I have had this problem.
I fixed it via this:
1.) Uninstall Origin (by the uninstaller provided in the Origin folder) {I also habitualy ended the origin.exe process in the taskmanager prior to this, don't know if thats relevent though)
2.) Downloaded a fresh installer from the origin website http://www.origin.com/download.
3.) Reinstalled Origin {I happen to have it installed on the same partition that ME3 is on, however this should not be relevent.}
4.) Magical voodoo (nah, I'm just kidding)
5.) Ran ME3 from Origin, didn't get the Origin (EA) is no longer running" error.
As a note, I did try out the solutions privided by EA (firewall issues), however this had no apparent effect (still had the error).
Additonaly, when I reinstalled Origin, it updated, might be what actualy fixed it. (It also complained that my cloud save was 1 min older than the save on my computer.)
Just ran into this problem. I didn't have to uninstall. I just ran the update and it worked. Thanks for the link.
