I have a stone fireplace w/o mortar and I managed to pound in a skinny nail between the stones.

If that doesn't work, they make some kind of clips used to hang stuff on brick; not sure what it's called.

They are called "brick clips". I have found them on ebay and thru catalogs. I used them on the exterior of my last house which they fit into the bricks easily.

Based on thier size I don't know if they would work the same for stone. They are pretty much sized specifically for bricks.

I never heard of the Brick Clips...they look neat. My parents have a huge brick fireplace and always used skinny nails to hang things.

Here is a link that might be useful: brick clips

I have tried these brick clips from Home Depot and they are meant for standard brick. It didn't work for my stacked stone fireplace with varring widths.

This mounts to the ceiling and you hang the picture from fine cables or clear fishline:

http://www.systematicart.com/home-office-solutions/slim-line-ceiling-mounted.html 

These posts put me I the mood to finally find a solution to hang my clock on the fireplace today. I have a dry stacked stone fireplace. I used a drill bit and drilled a small hole in between the bricks. You can't even see the hole when the screw is removed and it didn't damage my stone. Super easy!

I just Googled "picture rail ceiling-mounted" and it turned up a number of options (lazy_gardens's link didn't work for me, but I suspect it's the same idea). I have a traditional wooden picture rail in my 1911 house, and use fishing line to hang artwork using molding hooks. The ceiling mounted versions are engineered differently, but are basically the same idea.

I used the link which didn't work but searched the site and found this: https://www.systematicart.com/ceiling-mounted-coupler-set.html