[Sprint Samsung Galaxy S III] Tromatic:
Since I'm a "noob" and got mine working, I figure I'll lay out what I did.

I "hard-bricked" mine trying to install SuperNexus. It WAS the correct version, I have no idea what I did wrong. Since it was working for others, I can only assume a problem at my end. The phone was totally dead. Nothing, not even the charging LED. Windows would beep and buzz as I mashed various combinations of buttons, but that was it. Then it was this thread to the rescue!

I had to go get an SD card reader and a 16gb Micro SD card. I got the SanDisk 16gb, and specifically asked if it was a "Class 10". We had to Google the thing to make sure. I'm not sure the "Class 10" is really the issue, but it does seem to be a solution.

I followed the instructions from the poster on page three of this thread who used Cygwin. All you need to download and install is the "terminal emulator". Using the instructions on page three, I figured out what "dev" (Linux for disk) the Micro SD was and went on to flash the image. Unzip the de-brick file using 7Zip first. It took about three minutes to write the file to the card.

The phone had the battery in and was connected to the PC when I inserted the SD. Nothing, nada. No response to the power switch or button combination to get to "download" mode.

I decided to re-format the Micro SD card, but Windows 7 would not let me format the thing. My only option was as a 60mb drive. I downloaded a copy of EaseUS Partition Master (freeware) and re-formatted the card as a FAT32 disk. NOTE: If you have not used partitioning tools like that before, be VERY careful you select the right drive to work with. Be nice and calm, with no distractions, as you can do very bad things, instantly.

Flashed the card again, and repeated the process. I was prepared to have another failure. Inserted the card, mashed power+home+volume down buttons while muttering to the Android Gods for help. Bzzzz! The continue to download screen appeared! Just getting that far was awesome, as I no longer had a hard-brick. Volume up, and I was at the download screen.

I then fired up the one-click from Rwilco12's site. The progress bar on the app itself was showing prgress, but there was no progress bar on the phone, so I figured I was still dead. Took five minutes or so, and then the phone re-booted. SUCCESS! I set it up, then turned it off, removed the recovery SD card and inserted the regular SD card with TWRP backups and the rest. Crossed my fingers and turned it on. SUCCESS!

For those still having problems, I would suggest starting from the very beginning. Reformat the SD card to FAT32 using a partition tool and re-flash the recovery image. Follow the instructions carefully as written on page one (for Linux) or page three (for Cygwin).

NOTE: There is a difference between the two regarding the de-brick image. Unzip the de-brick file. The extension of the file you drag-and-drop should in fact be the unzipped *.img file, not the *.xz file. If you can get the thing to boot and get to the download screen, you should be golden. Once you start the restore process with Odin or the one-click file, it should not matter what the state of the file system on the phone itself is, as that file system gets formatted and overwritten right off the bat. Be SURE you are using the right recovery files. This is for the Sprint SPH-L710 only! Try re-installing the Samsung USB phone drivers, too. The "PIT file" problem makes no sense unless there is some communication error or the restore file is damaged or incorrect. Also, before trying to restore the phone, try a re-boot. Make sure there is nothing else trying to access the phone. If you have more than one instance of Odin running, a flash will usually fail.

I HOPE that makes sense.

EAT: Also found this regarding PIT file errors, may or may not apply the the SPH-L710:

Try changing the cable and the usb port you're using. If you are connected to a usb hub that will also cause major issues. Lastly make sure you remove Samsung kies before attempting to use Odin. They conflict with each other.