This can be a problem with Ubuntu/10.04 (lucid) in the user agent that isn't recognized by the site.

Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.9.2.12) Gecko/20101027 Ubuntu/10.04 (lucid) Firefox/3.6.12

You can try to set the general.useragent pref that adds the Ubuntu/10.04 (lucid) value to an empty string by deleting that text via the right-click context menu and Modify.
If Ubuntu/10.04 (lucid) Firefox/3.6.12 is one pref then sure to leave the Firefox/3.6.12 part.

That does not work. Neither does using "User Agent Switcher" to make FireFox respond as IE.

Be careful with setting the user agent to IE because that can make website send content (JavaScript and CSS) files that only work in IE and not in Firefox.
Google Docs should work in Firefox with the normal settings, unless Ubuntu made some changes with their branded version that is causing problems on that site.

• "Clear the Cache": Tools > Options > Advanced > Network > Offline Storage (Cache): "Clear Now"
• "Remove the Cookies" from sites that cause problems: Tools > Options > Privacy > Cookies: "Show Cookies"

Start Firefox in Diagnose Firefox issues using Troubleshoot Mode to check if one of your add-ons is causing your problem (switch to the DEFAULT theme: Tools > Add-ons > Themes).

• Don't make any changes on the Safe mode start window.

See Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems and Troubleshoot issues with plugins like Flash or Java to fix common Firefox problems

I am able to use my previous cookie to stay logged into Google Docs, but it keeps telling me that I have an incompatible browser after upgrading. I rebooted, even, and tried running firefox both as root and as myself

No change.