Someone sent me log samples from the DI-604 a couple of months ago, and they were used to add WallWatcher support for that router. That's also how I know that the DI-604 can send log records to Syslog port 514.
Before that, other people had told me they could not get SysLog style logging to work with the DI-604. This suggests one of two things: there's a newer model available that supports SysLog, while the older ones don't; or there's newer Firmware available that adds SysLog to models that couldn't do it with previous Firmware.
You may want to check D-Link's website to find out what's available for your model of DI-604.
Also, you may want to check Duane's suggestion that a software firewall on the computer running WallWatcher is blocking communication. Some software firewalls do that silently, without asking you whether the log records or WW should be allowed to use the 'net. The firewalls have Configuration sections that let you change the Permissions for specific applications, address, and ports. The Router should be in your "Trusted zone", and anything it sends to port 514 should be allowed to reach WW. Also, WW should be allowed to do DNS lookups (port 53) if you want to use the "Convert" options on its Logging menu.