I spent twelve hours today trying to root my new GS3. I registered to say thanks to everyone who has offered their input and to relay my experience for others.
I ordered the phone off Amazon last week and I activated it on Tuesday 8/22/2013.
This morning I tried rooting it with just Odin, some one-click method I found, and failed due to "can't mount sd card" error in recovery. Two subsequent rooting methods I tried failed with the same error message. I searched google and read forums for hours. I found some threads where people had similar problems with encrypted phones, but I verified mine isn't encrypted. (Thank goodness I decided to keep searching instead of trying to use Odin to push a factory reset. What a waste of time that would've been.) Then I found the thread where some devs had uploaded a few trial image files to push with adb, but I was unsuccessful there of course because I hadn't gained root access (su not found).
Finally I found a post in this forum from a guy who had precisely the same situation as myself, and he'd successfully rooted a new SG3 via the Samsung GS3 Toolkit. I tried that method and it worked.
I don't know why that worked and none of the others worked. I'm a super noob, so I have absolutely no insight. Maybe some of yall have some ideas as to why the toolkit method is the only way to gain root access on these newer S3's. Hopefully tutorials can be amended to include that information. It sure did take a lot of searching and reading for me to figure out! *whew*