root explorer not needed if just for that
If you've gone through the whole process of rooting your phone (look at me, almost sounding like I know what I'm talking about, and yet I got my first phone in about 7-8 years yesterday!!), you can use the same cmd shell to rename the file.
from the regular windows prompt, under the tools folder as previously stated:
adb shell
cd /system/app
mv I500_BingSearchAndroid_07152010.apk I500_BingSearchAndroid_07152010.apk.old
But I must point out that the whole process didn't work for me at first. In order for Bing to stop showing up, I also had to rename the following file as well:
/data/app/com.microsoft.mobileexperiences.bing.apk to old
Once I renamed that file, the Bing icon disapeared from the applications, and the search was finally using Google. The difference *may* be that I had updated Bing from the apps folder (prior to finding this thread), after cliking on it and being prompt to install the latest version.