Type about:config into the Firefox address bar. This will take you to the config warning page that says "This might void your warranty." Take this warning label with a grain of salt, since Firefox doesn't have a warranty. However, this is the advanced system settings area of Firefox, so don't go messing around with stuff unless you're a groovy expert.
With that said, Click the "I'll be careful, I promise!" button.
In the Filter box, Type in keyword.URL
Under Preference Name, keyworld.URL should pop up. Right-Click it and Select Modify.
The default URL is likely Bing, just erase it. Now all you need to do is Paste in the following URL that matches the search engine you would like the bar to use.
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Google Search: http://www.google.com/search?ie=UTF-8&oe=utf-8&q= Bing Search: http://www.bing.com/results.aspx?q= Yahoo! Search: http://search.yahoo.com/search?p=
You can also use a custom search engine of your choice, you'll just need to get the correct URL code for it.
Once entered, Click the OK button.
Changes should take effect immediately, to test it out open a new window (Ctrl+N) and type a search query into the address bar. If it performs a search on the search engine that you specified then you can close your about:config window and call it groovy. Now you're using a search engine that you prefer, and you won't be bothered by the default one.
In my example, I changed my Firefox 4 default search provider to Google. Bing has it's uses, but I find that it just doesn't compare to Google in terms of speedy, relevant, search results.