Here is a simplified diagram of the network in question:
[ASA 5505] <-> [Server 2008 R2 (DC/AD/DHCP/DNS)] <-> [Switch (Nortel Business Ethernet Switch 1020-48T-PWR)] <-> [Patch] <-> [Drop] <-> [Computer/devices]
The problem seems to be at the switch. If I bypass switch, things work well. I thought I had a bad switch, so I swapped it out, and had the same issue.
The Nortel switch is straight out of the box (I have three of them, all acting the same), so it seems that I need to do some config on them to get it working correctly, but I have no idea what to start with. Nortel is gone, so I can't get a firmware update or support for this.