I had the same problem. I just now ran the chmod command you listed, and the problem went away. Now, however, I have a new problem:

$ ./LimeWireLinux.bin
Preparing to install...
Extracting the installation resources from the installer archive...
Configuring the installer for this system's environment...
No Java virtual machine could be found from your PATH
environment variable. You must install a VM prior to
running this program.

The error is the same with sh LimeWireLinux.bin. I have successfully installed jre 1.3.1 (OpenOffice recognized it with some browsing). Now, my jre folder is actually in /home/[user]/jre 1.3.1 as that is where the folder installed to (can you tell I am new to Linux!).

Help greatly appreciated!!!