W5100 Ethernet Shield And Due

I started a thread about this on the Programming forum, but it's looking like it is likely a Due-specific problem, so thought someone here might have come across this problem before.

Here's what's happening:

Regardless of what firmware I use (including the standard Ethernet example sketches), the Ethernet will not connect to my PC more than about 1 in 4 restarts, if that. I've tried multiple PCs, Dues, cables (bth straight-through and crossover), shields, and example programs, and the result is always the same. Using the same shield, cable, and firmware, but replacing the Due with a Mega2560 gets it working perfectly every time. With the Due, it will connect, eventually, without me doing anything other than waiting. It may take a minute, or it may take an hour, but eventually it WILL connect, and it will work fine after that, until the Due is reset or power-cycled.

Anyone have any idea what's going on here?

Regards,
Ray L.