[Sprint Samsung Galaxy S III] DTDMike:
I've found a sure fire solution(Only if the toolkit was used)

Ok, I had this exact problem with my SGH-T999(T-Mobile S III), I was on 4.1.2, & used the toolkit 4.1.1 method. Next thing I knew my S3's Wifi would not turn on. Here's how I fixed it. It turns out that In the process of rooting, when you install the insecure boot on your phone, that will cause the wifi to not work. So what we want to do is make a backup of our boot.img, before hand. then restore it after the root process is complete.

If your phone is model SGH-T999 then you're in luck because you don't have to do most of this stuff, as I've already done it for you. Just Skip to the part that says "SGH-T999 4.1.2 ONLY!"

What you need:

1.) Your Galaxy S III(Not rooted)

2.) The Odin flashable CWM Recovery for your specific model

3.) Odin 3.07

Pre-fix Step.) If your phone has already undergone the root process, then you're going to need to restore the stock rom, unless you've already made a nandroid backup somehow.

Steps:

1.) (Skip to step 5 if you already have CWM Recovery Installed) Boot your phone into download mode, by powering it off then, holding VolDown+Home+Power. Once you see the prompt press VolUp, then plug your phone into your PC & leave it like that.

2.) On your PC open Odin 3.07, make sure only, Auto Reboot, & F. Reset Time are checked.

3.) Now, in Odin, click the PDA button, & navigate to your SIII Toolkit folder, once there open the recovery folder, there should be a bunch of tar files, choose the appropriate one(if your SIII is a model SGH-T999 then the file name will be something like this "recovery-cwmtouch-6.0.2.3-SGHT999.tar")

4.) After choosing the appropriate file, click start on odin, once your phone reboots you'll then have CWM Recovery on your phone.

5.) (Skip to step 7 if you already have a recent backup) Now we want to boot our phone into the new recovery mode by powering it off, then holding VolUp+Home+Power

6.) Once your phone boots into CWM Recovery, Goto "Backup and Restore", then click "Backup", proceed with the backup process, & then boot your phone normally.

7.) Now plug the phone into your PC, and navigate to the Internal memory, or external if your backed up to External SD, open the "clockworkmod" folder then the "backup" folder. Now open the folder that contains the backup that we made.) and copy the "boot.img" file to your PC where you can find it later.

8.) (Skip If you already had a backup, & have already done the root via the toolkit.) Now go ahead and follow the toolkit's rooting instructions.

9.) Now your phone is rooted, but the Wi-fi Likely won't turn on. (If it does your lucky)

10.) Now it's time to restore the original boot partition. First we want to turn that "boot.img" into "boot.tar.md5". To do that follow the tutorial here(Replacing the parts that say recovery with boot) "w0lfdroid .com/2013/04/android-how-to-convert-img-recovery-to.html"(Remove spaces)

11.) When you're done, there should be a file named "boot.tar.md5" in "C:\cygwin\home\%USER%"(%USER% is where your username should go.) If it's there your almost done, if not then you need to repeat step 10 thoroughly.

12.) Now the final step is to repeat steps 1-4, but in step three this time we want to navigate to our newly created boot.tar.md5, Instead of going to the toolkit recovery folder. After you complete these steps(AGAIN, using the boot.tar.md5, & not "recovery-cwmtouch-6.0.2.3-xxxxxxx.tar") your Wi-Fi should now work again.

SGH-T999 4.1.2 ONLY! (Any other model will be bricked by using this method):

Since you've got the same Model Phone, as well as the same Android version as me, you can take quite the shortcut. Reducing it down to only a few steps. Now Lets Start.

What you need:

1.) Your Galaxy S III(SGH-T999, with Android 4.1.2) that is experiencing this problem after rooting via toolkit.

2.) This file "https ://dl.dropboxusercontent .com/u/36653831/boot.tar.md5"(Remove Spaces)

3.) Odin 3.07

Steps:

1.) Boot your phone into download mode, by powering it off then, holding VolDown+Home+Power. Once you see the prompt press VolUp, then plug your phone into your PC & leave it like that.

2.) On your PC open Odin 3.07, make sure only, Auto Reboot, & F. Reset Time are checked.

3.) Now, in Odin, click the PDA button, & navigate to the "boot.tar.md5" you downloaded, select it, then click start. Once your phone restarts your wifi should work just fine.

Side Notes:

Let me know if this works for you.

If you use my boot.tar.md5 the you'll be getting the stock secure boot for you phone.

Note for Admins: You might want to sticky this post if people find it helpful.