I FOUND A SOLUTION!!!
I was having the same problem on my Samsung Galaxy S3 SGH-I747M (This may work for I9300 as well, but I'm not sure since I don't own one).
I tried reformatting using the phone - didn't work. I tried reformatting to FAT32 and NTFS on my Win7 laptop. Didn't work. Every time, it would work for a couple of minutes and let me take some photos, and then it would stop working as soon as I tried to take a video or tried to go into the file browser and create folders.
So...
Here's the fix for SD Card on Samsung Galaxy S3 not writable (or at least what worked for me), using a Windows PC (probably Vista or later):
--- 1) format the SD card to "exFAT" (it's not necessary to do a full format - a quick format will suffice)
--- 2) "safely" eject the SD card by:
---------- 2a) using the "Safely Remove Hardware and Eject Media" icon in the system tray, or by
---------- 2b) opening "My Computer" and right-clicking the drive and selecting "Eject"
--- 3) remove the card from your PC and place it into your GS3
--- 4) give it 10-15 seconds to prepare the card
--- 5) Voila! C'est bien! It should work for you, now.
Hope this helps some of you frustrated peeps out there - I was getting pretty frustrated, since I bought my SD card from The Source by Circuit City and their return policy is not the most friendly...
P.S. It's still working for me, even now!
If it doesn't work for you, I'm sorry this didn't help and good luck as you continue the search. But please don't abandon Android for Apple - you KNOW that's a mistake, as Apple doesn't even support external storage, and they're really annoying (and a lot like Microsoft in its developmental years - LITTLE CUSTOMIZATION without using work arounds (like jailbreaking) and slow progress to reverse that trend!)
Cheers,
Nalorin