Nerd Alert Ultra-specific nerdery.

32-bit Java Runtime Environment needs to be installed (which can be done through synpatic/apt-get).

Run the following at the command line:

export PATH=/usr/lib/jvm/ia32-java-6-sun/bin/:$PATH

Install the firefox-2 package from synaptic/apt-get.

Start the Flex installer:

chmod a+x flexbuilder_linux_install_a3_033108.bin
./flexbuilder_linux_install_a3_033108.bin

At the end of the install process, it may explain that you need to manually copy a file to the Firefox 2 plugins directory:

sudo cp Adobe_Flex_Builder_Linux/Player/linux/install_flash_player_9_linux/libflashplayer.so /usr/lib/firefox/plugins/

Start Eclipse and the Flex Builder 3 Plugin should be available.