6 years ago we installed radiant floor heating in our kitchen.  We live in an old house and our kitchen used to be a porch.  This means there is very little foundation or support. 
When we gutted the kitchen we found that it was supported by a limestone foundation and two 8x8 railroad ties.  We reinforced the floor with additional floor joists.  There is no basement under this section of the house.  We also added 2 inch blue board between each joist and calked them in for added insulation. 
Layer 1 - Subfloor
Layer 2 - tubing, sand and nailers
Layer 3 - plywood subfloor
Layer 4 - concrete backer board
layer 5 tile

About two years into the project we noticed our tiles started popping and coming loose....we have now removed all tile and have re-gutted the kitchen.  After removing the tile and concrete backer board, we noticed that the "layer 3" sub floor was black.  Now after tearing it up we see that it is covered in BLACK mold! 

We have removed all base cabinets and noticed some mold at the base of the floor here too. 

What is causing this and how do we prevent it from happening?

We need to have radiant floor heat in this room because when we remolded we removed the fintube heating system that was in place. 

Any suggestions would be appreciated. 

is it outdoor temperature below the blue board? i.e. in the foundation, it's pretty much as cold as it is outdoors?

where are you?

I am suspecting that isn't nearly insulation and you may be condensing moisture in your floor assembly. but you say limestone foundation... that usually means wet ground. you could also have prevalent moisture issues from that, if it hasn't been addressed properly. you might have even sealed in mold from the beginning.

Suffice to say you'll need an expert in mold mitigation for this one, not radiant heat guys.

Radiant floor heating cannot cause mold unless the system is leaking and re-filled. Your problem lies elsewhere...water vapor trapped between layers. Look to the source of moisture and the presence and location of the barrier, if any.

I know someone who sealed up a crawlspace and then found all the joists rotting away from mold. But layer 3 and not layer 1 - maybe liquids spilled on the floor?

Is the house air conditioned?

Got a zip code/location (for weather data purposes)