Gregory,
You cannot PING an interface on the ASA if you're not on that same interface.
In other words...
From the inside can only PING the inside interface
From the outside can only PING the outside interface and so on...
Now, you should be able to PING from a DMZ device to an inside device for example.
Since the DMZ has a security level of 50 and the inside of 100, you need a STATIC NAT and and ACL.
static (inside,DMZ) 10.10.4.0 10.10.4.0 netmask 255.255.255.0
access-list DMZ permit ip 10.10.8.0 255.255.255.0 10.10.4.0 255.255.255.0
access-group DMZ in interface DMZ
Federico.