Update: Upgraded to 1.3.7 (not sure why I was running 1.3.6) and still having the same problem with the Linux client as well as a Windows 7 client. I read that someone was having a problem with their wireless router and my network guy said I should use a switch instead which I changed this morning, but it has had no effect on the problem.
I am running 1.3.6 from a 32-bit Windows XP as a server and 1.3.6 from a 32-bit Centos 6 client. My client continuously loses the connection with 'server is dead' message. The client will then reconnect, get the server is dead message, and on and on. Below are the logs showing it connect and then the connect-disconnect cycle. Even though the logs show that the client isn't able to reconnect, moving the mouse back to the client works fine until the 'server is dead' message which happens every 7-25 seconds.
Synergy 1.3.6 Client on Linux 2.6.32-71.29.1.el6.i686 #1 SMP Mon Jun 27 18:07:00 BST 2011 i686 XOpenDisplay(":0.0") .... .... connecting to 'Main': 192.168.1.148:24800 clipboard blah blah.. connected to server entering screen clipboard blah blah.. server is dead leaving screen failed to connect to server: server is not responding retry in 1 seconds connecting to 'Main': 192.168.1.148:24800 connected to server server is dead failed to connect to server: server is not responding