Is it possible to give a texture thickness? I have a plane with a texture with both transparent and opaque elements. I would like to give that texture thickness. If I solidify the plane, the texture is only on the faces, not 'through' the solidified plane. I could apply array modifier in very tiny increments to build the thickness, but that's not optimal and seems not really smart either.

UPDATE: More details:

My initial request for this project:
Is it possible to convert image texture's black color to transparent?

I am trying to recreate series of confocal microscopic images into 3D structures. Each image is flat, I tried to give them thickness by adding array of 6 planes but it looks awful. If I could give each plane thickness, they would stack in nice 3D shapes. 



UPDATE 2
David - I tried your suggestion, but I'm getting no effect after displacement. I added solidify modifier on top of that, but I only get fill on edges / rims:



FINAL UPDATE:
I followed atomicbezierslinger method - it gave me pretty decent results with very good 3D reconstruction. I used 100x100 cubes per picture, resized z-size to 0.2. Attached is a render of 13 z-stacks from confocal. I need to tweak noise a bit, but I guess there's no better way to go.
Now I need to write a Python code to automate it - each data set is 100 z-stacks - too much to do it by hand.
I appreciate all suggestions and your help.



FOLLOW UP:
I probably was unclear asking initial question and didn't explain my goal well. Confocal microscope uses laser to scan fluorescent-labeled animal tissue. Each laser pass reads a plane of ~1 micron thickness then, the next scan goes deeper and deeper. This way z-stacks are created. There is a software for 3D reconstruction of z-stacks (e.g. ImageJ) but I am exploring possibilities and limitations of Blender in this application. Please see below image for the explanation. 

Picture source: Hammad et al., Arch Toxicol (2014) 88:1161–1183
Voxels

Consider some images where the planes (rectangles) are either parallel or pependicular .. based on cubes.  Flat Surface on Top.


Small Sample of Color Transparency Threshold





Small Sample of High Density Mesh from distorted image from user provided image from natural light camera. Image below.





Sample from artificial image with more precise edges. Image below.




Steps


Create a high density mesh with cubes.  Simple Cube (6 sides) with the 2 array modifiers, vertical Z (1) and horizontal X (1), repeat 40 or more on both, then apply both. (I suggest user use a high number 500 or more if possible)Vertex Remove Doubles. See image above.   The number 40 was for a quick test.
UV Project . Turn specularity to zero. Add your texture. Affect color and transparency.
Render


Observations


High Density Mesh. Sheet of cubes. Geometry modified. One mesh of many cubes internally. High Density Mesh is the same requirement for Displacement in texture panel. See below.
Please Improve. I used a visible margin for each cube in the original array modifier, in the example below.
You can have zero margin.  Thus some lines in interior.  
The more intricate the transparency the more need for density with this technique.
The texture below has .png transparency.  Careful image processing is necessary and the best image, or least offensive image, here can be improved.


The perceived thickness is determined by the alpha of the channel. Your can change transparent image or image sequence. with no further work. No particular modelling.  There are other ways to do this.  Quick example.  I have added an array modifier only to show you two views at once. Front and Side. Any undesirable details have not been investigated.



Displacement method.  Dense mesh from a plane subdivided. Texture affecting displacement.  There is no flat surface.  This technique is documented many places.  This may be the least suited to your purposes. Consider Subdivision Modifier. The image also shows the triangulated surface just for discussion purposes.



All the above should be adjusted to you tastes.

Optionally consider a particle system with the 


Texture determining [density].  
Texture determining [color] in emitter.