I own an S3 since December. When I bought it I also bought a 32GB SD card. I've been using it without any problem until about 2 weeks ago. At that time the phone started complaining about the SD card being unformatted etc or failed to mount it at all.
I read about this problem in various threads. Most of the people complained but none proposed a valid solution. All except for stas333 who gave this answer:
http://forums.androidcentral.com/sprint-galaxy-s-iii/240489-sd-card-unmounting-itself.html#post2633130
It is not a software problem (it would have been corrected by Samsung till now) nor a problem of the SD card (mine works when I put it in another phone or the PC). So, either the SDHC i/f in S3 has a bug or the SDHC connector is the source of the problem. Since I could not do anything about the SDHC i/f (after all it worked until 2 weeks ago and it still does occasionally), I tried to "fix" the connector. stas333 suggests you put a sticky note on the card and place it inside the slot. I used blanco correction fluid and put some of it on the top side of the SD card, at the part which is above the pins (that lie on the bottom side). I waited for it to dry and then put some more. After a few minutes I used a tape (sellotape) to cover the top side of the SD card (and the area that I had put blanco on). The purpose was to make sure that blanco was not turned into pieces when the SD card was placed into the slot. Then I put the SD card inside the slot (with care...). So far it works.
There is high probability that the SDHC connector is the source of the problem. It explains why this happens to some SD cards but not all (maybe the "good" ones are a bit thicker). It also explains why some cards are eventually destroyed. Since there is no good connectivity between the SD card's pins and the connector's pins, the card is destroyed when i.e. the power pin does not have good contact and thus the power to the card is switched on and off too quickly.
Well, in any case the procedure is quite safe and you've nothing to lose. It would be better to switch the phone off before trying this out. Be careful when you slide the SD card back into the slot so that whatever you put on top of it to make it thicker remains in place!