Dogecoin daemon (dogecoind) not responding to RPC requests
Figured it out.  It had to do with adding rpcallowip lines to the config file.  Adding these three lines solved it:

rpcallowip=127.0.0.1
rpcallowip=192.168.1.*
rpcallowip=0.0.0.0


The last line is what eventually allowed it to work.  I'm not sure why, but netstat said it was listening on 0.0.0.0:22556, so adding 0.0.0.0 to the rpcallowip list solved it.