It may simply be the Kindle power management daemon that's doing its thing... (AFAIK, there's no rules blocking ICMP pings, ever, so, iptables probably has nothing much to do with this
Do an open ssh connection actually hang? (EDIT: reread your message, yes. That's weird).
(And, yeah, you could look at /var/log/messages, dmesg, or a full ;dumpMessages).
Just to be sure, do the wifi actually work on the Kindle during these 'blackouts'? (ie. the browser).