All,
Many thanks to everyone who commented on this issue. I believe that I have solved it.
It turns out that the number of nfsd's that I was running (32) was way too low. I observed that adding more nfsd's when NFS was hung always caused the hang to go away immediately. Now I am in the tuning stage where I'm adding more nfsd's until there are no more hangs. I am up to 172 of them now, and the hang frequency has decreased by about a factor of six. Evidently my workload has changed when I wasn't looking closely enough. I'll probably end up with about 256 nfsd's.
For the sake of completeness, here's how to change the number of nfsd's on the fly:
echo 172 > /proc/fs/nfsd/threads
and, of course, edit /etc/sysconfig/nfs to change RPCNFSDCOUNT to set the value for the next boot.
Steve