I am trying to fasten furring strips to my brick fireplace. I would like to know if using a construction ahesive and a few well placed hammer nails will work or do I have to drill and anchor every furring strip! (Please No!) What's the best way to do this?

Yes construction adhesive and a few well placed cut nails should do it.

Should hold it yes... Not always! The easiest way to do it and know for sure it will stay put is anchor it with tapcons.
Mike

I'm liking Mike's idea of the glue and Tapcons. Not a big fan of cut nails myself. It doesn't seem that you will need a huge amount of strips (maybe 3 horiz. Rows?) so it shouldn't be too tough. Get a hammer drill though! I usually go into the mortar joints, just in case somebody decides to undo your beautiful work 20 years from now. Why compromise the brick if it's not absolutely necessary, right?

Art, you said you prefer to screw or nail into the mortar just for future purposes. Is that because if that day ever comes, when the screws are removed the mortar can be patched?

I am trying to fasten furring strips to my brick fireplace. I would like to know if using a construction ahesive and a few well placed hammer nails will work or do I have to drill and anchor every furring strip! (Please No!) Whats the best way to do this?
If this is a active fireplace you might want to check with local building codes to see if you can install a wood product that close to a open flame.

Art, you said you prefer to screw or nail into the mortar just for future purposes. Is that because if that day ever comes, when the screws are removed the mortar can be patched?

Correct.