I've earned my "tumbleweed" badge as a result of this one. pats self on back
What I've learned:
My solution:
1) Removed the left hard drive from the dns-323, which is half of the RAID1 volume.
2) Mounted (ext3) this drive using a USB-to-SATA adapter to the machine where I run rsync.
3) Performed the rsync command for the restore outlined above. I removed the --delete option which really shouldn't be there and I added the option --size-only. The size-only option made it so that timestamps were essentially the only thing that got restored, since files had already restored properly.
4) Unmounted the left drive from the Atom machine and returned that drive to the dns-323, while also removing the right drive. The right drive needs to be removed so that the dns-323 recognizes that the RAID volume is degraded.
5) Re-add the right drive to the dns-323 and tell it to rebuild the RAID volume.
6) All timestamps are now good.
A possible alternate solution:
I've read enough about rsync and NFS/Samba/cifs now to understand that this problem is likely related to permissions on the NFS server (dns-323). Internally, the user/group ids in the dns-323 are 501/501. No permutation of how I mounted the dns-323 on the Atom box would allow rsync to properly set timestamps. I do believe that changing my user account on the Atom box to have uid/gid of 501/501 would have worked, though. My user had the default 1000/1000 and root had 0/0 IIRC.