Back in previous versions of LMDE you could right click the .jar file and open with Java6 Runtime. I don't seem to be able to do that in the latest version, Java Runtime is not an option.
In LinuxMint 12 you can open .jar files by left double click and Archive Manager will open it. To run the .jar file you need the Sun Java 6 Runtime files which, if installed you can select via a right click from the drop down menu. To set the icon path you can use "java -jar /home/your name/directory/filename.jar"
The jre files that are installed on my system are:
sun-java6-bin sun-java6-jre sun-java6-plugin
"Open With Sun Java 6 Runtime" does not appear as an association choice on right click but it does appear as an option when a .jar file is right clicked.