Dec 18 17:06:27 10.0.0.1 kernel: NET: 14 messages suppressed. Dec 18 17:06:27 10.0.0.1 kernel: ip_conntrack: table full, dropping packet.
A-ha! Some Google searching led me to this Wiki article on router slowdown.
The Linksys WRT54G/DD-WRT firmware defaults of 512 connections (too low!) and 3600 second timeouts (too high) were obviously causing the problems. I bumped up the connections to 2048 and dropped the timeouts to 1800. Hopefully it'll fix the problem.