1. Download the LimewireLinux.rpm file.

2. Make sure you have "alien" installed:
sudo apt-get install alien

3. Create a debian package from the rpm:
sudo alien LimewireLinux.rpm

4. Install the debian package:
sudo dpkg -i limewire-free_{version-number}_i386.deb

I do that, but when I go to application and try to run Limewire, nothing happens.

Here's a way to combine steps 3 and 4 without creating an intermediate .deb file:

sudo alien -i LimewireLinux.rpm


-Tim