My stock EH03 Fascinate is rooted. It isn't hard to do. I used ODIN and CWM Recovery to do it.
You need ODIN on your computer. CWM-Recovery-All-3-30-fix.tar on your computer where you know where it is and su-2.3.6.1+busybox1.19zip on your SD card in your phone.
Use ODIN to flash the CWM-Recovery file. (If you are new to ODIN read the instructions first!)
Put the battery back in but do not reboot yet.
With the phone powered off, press and hold the Volume UP, Volume Down and Power button simultaneously. Hold the buttons until the Samsung logo comes up, goes black, then comes back up. Release the buttons now and you'll enter CWM.
Scroll down to "install zip from SD card" and hit the Power button.
Scroll down to :choose zip from SD card" and hit Power button.
Scroll down to the su-2.3.6.1+busybox1.19.zip file and hit the Power button.
After the file has finished installing you can hit the menu button to back out to the main Clockwork Recovery menu. Do not reboot your phone at this point IF you want to flash a ROM or Kernel. Rebooting your phone will replace the CWM we flashed with the stock recovery. If you just wanted to root your phone you can now reboot it.
I do not take credit for these instructions. They were given to me by several helpful people on the forum. I am just passing the information along. If you are not familar with ODIN, please read the instructions first before using the program and follow them exactly. A search on ODIN will find them.
Note: once your phone is rooted it will no longer receive the OTA updates. To get them you will either have to flash a custom ROM or unroot the phone.