Wine - and its commercial cousin CodeWeavers' CrossOver Office allow Windows programs to run under Linux. Depending on the program, this works remarkably well.
It would be nice if OriginLab would collaborate with CodeWeavers to ensure that Origin works flawlessly under WINE and CrossOver. The most visible way to do so is to participate at CodeWeavers' certification program: http://www.codeweavers.com/site/compatibility/certify/
But even if OriginLab would only test Origin under WINE from time to time, it would be a great way forward!