No version of the GS3 requires you to use a SD card to install a custom ROm of recovery mode, it just makes it a lot easier.
If you're running 4.0.x - 4.2.x, you're fine, and can install any ROM you please, with certain limitations (I don't think 4.2.x supports Cyanogenmod 9 or above, for example).
If you're running 4.3.x or above, your bootloader is locked, and there is, as of now, no way around it (however, there are a couple companies that can offer a downgrade to 4.1.2 with a downgraded bootloader, with mixed results, according to those who have used them), so there is NO way of installing a ROM that is not Touchwiz based.
If you're running 4.0.x - 4.2.x, here's how you do it:
1. Download a custom recovery mode (TWRP and Clockworkmod recovery are the two most popular, though, I prefer CWM, but the choice is yours), download whatever ROM you wanna stick on the device, and download the gapps file (for AOSP based OS's) for the version of Android you want on the device.
2. Hook your phone to your computer and run Odin (make sure Odin recognizes the phone is there).
3. Click on, "AP," and select whatever custom recovery mode you've chosen. Odin will install it to the phone.
4. When the new recovery mode is installed, power down the phone, and enter recovery mode (Go into recovery mode by pressing the volume up button, the home button, and the power button all at once, however, when some blue text appears on the top left of the display, release the power button. If the install was successful, it will enter the recovery mode you chose, if not, try it again, or look for solutions, I never had any problems with it, so I don't know what may cause it. Sorry bro). Make a full nandroid backup as well (just backup the phone).
5. Once you've made sure that the recovery mode was successfully installed, and a full backup of the OS you have is made, reboot the device.
6. With the phone still connected to the computer, create a new folder in the SD card (label it whatever the name of the OS you're installing is (ex. Cyanogenmod11) and move the ROM and gapps files to that folder on the card.
7. Power down the phone, and enter recovery mode again.
8. Wipe the cache for the device, as well as the dalvik cache.
9. Install the ROM and gapps files from the SD card.
10. Reboot, and enjoy your new ROM.
NOTE: This was how I did it two years ago on my AT&T GS3, and I'm not sure if that's still how you do it. I did it on 4.1.2, before the bootloader lockdown.
Good luck.