So, I had installed Mass Effect 3 on my Windows Vista laptop and also on my Wine Linux box and I decided to use Origin cloud to sync up everything so I could play on the go on the laptop and at the Linux box when I'm around. On Vista, it picked up my Mass Effect 3 saves from Wine, no problem. Apparently, I had issues when I was syncing my saves back to the Wine box. At first, I simply chalked it up to Wine's own deficiencies and moved on. When I went around to copying my saves from the Vista laptop to my Wine box however, I noticed that all the files touched by Origin's cloud save on Wine had 000 file permissions. I also checked that the files were of the same size as the Vista files I copied.
I infer that this means the cloud sync actually worked but only the file permissions were goofed up. Next time, I will try to simply chmod this. As an additional side note, cloud syncing on Wine only works when launching Mass Effect 3 since Origin in Wine cannot detect when the game is finished running to sync back the progress. The only way around this is to exit Origin, then restart again.