What were you actually doing with your Phone? Did you try to install custom ROM/Recovery/Kernel/Mod and so on?
I too have S3 GT-i9300 which is working fine after replacing my motherboard because I came across Sudden Death Syndrome (SDS) in S3.
* If you are able to go to download mode then download the latest Stock version of S3 which is 4.3 JellyBean and flash it via latest odin.
* Make sure USB Debugging is ON in your phone and your phone driver must be installed correctly.
* You may use [this link](http://developer.samsung.com/technical-doc/view.do?v=T000000117) to download driver for Samsung Mobile.
* Make sure before installing this software all the other previous/old driver are uninstalled.
* I would recommend that you should try to ADB to see your phone has proper drivers installed. Just Google ADB and download it. (Put all the files in one folder named `adb`.)
* Press +<kbd>R</kbd> to open **Run**. In it type `cmd`. Now **Command Prompt** should have started. (I assume that you put the folder `adb` in root of `C:` drive) Type the following commands.
1. `cd <space> /` - this will take you to root of `C:` drive
2. `cd adb` - this will take you in `adb` folder.
3. `adb devices` - this command will show all the devices currently attached to your system.
If you can't find any device then it means drivers are not properly installed.
Try it, hope this may help you.