Summing up from the comments above...
As eldarerathis wrote: "connected as a media device" suggests it's using MTP, which would require special software to be installed (there are MTP packages available for different Linux flavours).
Try getting UMS working
You might also want to check your "notification area" when the device is connected: I don't know the Kindle Fire, but it might offer to switch to UMS (USB mass storage) mode. If you don't find anything corresponding in the notifications, also take a look through the settings. Just a possibility, nothing granted.
Try it with MPT utilizing the MTP file system
If you cannot get UMS working, there's a helpful link pointed out by eldarerathis is Connecting Google Nexus 7 to Linux on how to make mtpfs working on Linux. Basically, this can be done by issuing 4 commands in a terminal:
sudo apt-get install mtpfs
Short explanation on what it does:
- install the
mtpfs package, which provides the MTP file system
- creating a directory where to mount your Kindle to
- actually mounting your device
- opening nautilus to show the files contained
Of course, the last step can be replaced by opening any file explorer you like :) And as the "missing sudo" on lines 2-4 indicates,
mtpfs is running in userspace.
Graphical MTP frontend
There's also a graphical MTP frontent available in the Ubuntu repositories: gMTP. It's mainly intended to be a MP3 player, but you also can manage your files with it:
sudo apt-get install gmtp
(it's in the universe branch, so you might have to enable universe first in your package manager)
(screenshots taken from LinuxAndLife)
So you could use this instead of Nautilus -- or next to Nautilus, depending on what you want to do. With gMTP installed, simply connect your Kindle via its USB cable, then launch gMTP. Click the Connect button then will mount the device, and the Connect button gets re-labeled to "Disconnect", as shown in above screenshots.