On Ubuntu, install the libreoffice-java-common package by calling this command in a terminal:
sudo apt-get install libreoffice-java-common
One problem solved by this is getting a long error message with "NoClassDefFoundError" during installation.On Ubuntu, if you get a message similar to Exception in thread "Thread-402 java.awt.HeadlessException" in LibreOffice/OpenOffice, see this stackoverflow answer. Note that the message might not appear in a dialog but only on the command line, so you might want to start LibreOffice/OpenOffice from a terminal window.
If you start LibreOffice/OpenOffice from the command line, and you get a NoClassDefFoundError, make sure you're not starting LibreOffice/OpenOffice from the LanguageTool installation directory.
Linux-specific issues
Updated over a week ago