OMG I have the EXACT same issue and I just registered and was about to post the same thing! I was following the guide in the top of this forum and the one at [Complete Guide] How To Fully Back Up And Restore Your Android Phone Using Nandroid Backup
I did the root method and got a super user on the phone. And then I downloaded Rom Manager from the market.
I launched it and I clicked on the Flash ClockworkMod Recovery and then picked Morotorla Droid 3, gave it Super User and then wait for it to finish.
Then I click on Backup Current Rom and it just reboots my phone. It doesn't go to any command line style screens it just goes straight to my homescreen just like is happening for you! I check SD card and my phone and my backup doesn't exist.
UPDATE:
Okay I figured out how to get past this.

So here is the steps, RyanGSams. The sticky thread mentions a "bootstrapper" but us beginners have no idea that we have to install that seperatley. So...
(You must have the ROM Manager installed already and you have clicked on the first button in that app, clicked Droid 3 and had that do its thing)
1. First download this apk file from here
http://www.droid-life.com/2011/09/16/droid-3-bootstrap-released-get-your-recovery-and-backups-on/
2. Make sure your phone is set to allow applications from Unknown sources
3. Transfer that file to your SD memory card in your phone
4. Disconnect from your computer
5. Use the file manager or other app you have installed to find the apk file you just put on your SD card. Open it. (If you didnt allow apps from unknown sources it won't work here)
6. Allow it to install. then Click on the first button
7. Allow it to have super user and wait for a "sucess!" message.
8. Click the second button which is "reboot into recovery"
9. Your phone will reboot and go to a boring dos like screen. Use the volume dpwn button to go to the option for backup/restore. Then open your keyboard and use the "return" key to select it
10. The phone will go about doing stuff for a while. It is making its backup. Mine is still going as I write this (about 4 minutes now). Will edit with anything else as I proceed...
So excited! I found the answer here
http://www.droidforums.net/forum/te...ns/113817-cant-backup-rom-manager-ascend.html to give credit where its due
Maybe these steps could be elaborated into the sticky thread? I know thats the first thing I followed and it says nothing in depth about the "boot strapper" thing. WHat is a bootstrapper anyway?