[Galaxy S III Themes and Apps  (US Carriers)] AndroidFan4:
This thread shows what AOSP (pure Google ICS) apps are safe to replace Touchwiz Apps with on your Galaxy S III to get that Nexus/CyanogenMod type pure ICS look and feel (see screenshots below). These have been tested with S-Voice and it is still able to open links and commands in the ICS apps.

I am not responsible for any issues or problems with your phone. You are messing with System Apps. I highly recommend creating a Nandroid Backup before starting in case something goes wrong

Requirements

Your phone must be rooted and have Super User installed. I highly recommend making a Nandroid Backup in ClockworkMod Recovery before moving system apps just in case.

Install Root Explorer and Titanium Backup from the Play Store.

Steps

Download the APK file you want (linked below)

Uninstall the original Touchwiz application through Titanium Backup then reboot your phone. I recommend backing up applications before you uninstall just in case something goes wrong or you want to restore them.

Open Root Explorer and navigate to your download folder under SD Card

Long press on your downloaded APK file, select move and navigate to your System App folder.

Select Mount R/W at the top of Root Explorer and touch Paste.

Long press on the APK and set permissions to rw-r-r-- (See image at end of post.)

Touch the moved APK file and install it. If you get an error message, make sure the Touchwiz Version of the app is uninstalled.

Reboot

, then try again.

Tested and confirmed to work

Gallery

Update 6/24: Uninstalling the Touchwiz Gallery causes the Touchwiz Camera to force close after taking a picture and touching the thumbnail preview on the bottom right side. Workaround: Close the camera app and open the AOSP Gallery to preview the picture.

Browser (Must first uninstall the Touchwiz Browser. It is listed as "[BOOKMARKS] Internet 4.0.4-T999UVALEM" in Titanium Backup. Reboot. Then proceed to step 5.)

Calendar (Must first uninstall the Touchwiz Calendar. It is listed as "Calendar 4.0.4-T999UVALEM" in Titanium Backup. Reboot. Then Proceed to step 5)

Update 6/25: Make sure you uninstall only the Touchwiz Calendar (you can tell by the icon listed in Titanium Backup.) If you uninstall the one with the AOSP Calendar Icon you may lose Calendar Sync and S-Voice support for the Calendar.

Messaging (Must first uninstall the Touchwiz MMS App. It is called "[SMS/MMS Prefs] M... 4.0.4-T999UVALEM" in Titanium Backup. Reboot. Then Proceed to step 5.)

Calculator

Movie Studio

(Update 6/29) Email/Exchange (Must install both AOSP Email and Exchange APK's in order to work. In order to install you Must first uninstall the Touchwiz Email AND Exchange App in Titanium Backup. Reboot. Then Proceed to step 5. I was having issues connecting to Hotmail/MSN/Live account servers but had no issues with all others. May be temporary server issue. More testing needed.)

Tested but had issues

Clock/Alarm (Force closes upon reboot)

Camera (Will not work. Thanks naturefreak85 for testing)

Still need to download and test

Phone/Dialer (isn't supposed to work due to Touchwiz frameworks)

Contacts (isn't supposed to work due to Touchwiz frameworks)

Grabbed from the Play Store

Nova Launcher

ICS Keyboard

Having Issues?

Read the OP completely

Make sure permissions are set properly on the app you're trying to install. In Root Explorer, long press on the APK you're trying to install and set permissions. It should be set like this:

Make sure the Touchwiz version of the app is uninstalled. Don't uninstall more than you need to (Calendar).

Reboot after making changes

Restore the original app using Titanium Backup or do a Nandroid restore if all else fails.

These files were pulled from various threads (like this & this) across XDA using the handy search tool. Credit goes to the community for providing instructions and files.