Using anatomical terms for human organs and parts of plants
First, for reference, see here for a discussion about the difference in directional terms between bipeds and quadrupeds as well as a fairly complete explanation of word meanings/etymology. 


The etymological meanings of the various anatomical directional terms should help explain their usage in body organs. For example:

Ventral -&gt; "belly" side
Dorsal -&gt; "back" side
Anterior -&gt; "before" or "toward the front"
Posterior -&gt; "after" or coming after (opposite to) the anterior 



For body organs (or other body parts):

This SE question addresses hands and feet, while this page from Bryn Mawr College directly demonstrates the usage of anatomical positional terms of the brain:




Rostral/ Anterior ---&gt;   Head or front end
Caudal/ Posterior ---&gt;  Tail or hind end
Dorsal  ---&gt;    Back or top side
Ventral ---&gt;    Belly or bottom side



Anatomical positional terminology for brains is consistent across organisms. Here is another picture demonstrating anatomical positional terminology of this organ from Santa Monica college:





For plants, see "Plant Identification Terminology: An Illustrated Glossary" (Harris &amp; Harris, 2001):


  Dorsal. Pertaining to the back or outward surface of an organ in relation to the axis, as in the lower surface of a leaf; abaxial. Figure 407.
  
  Ventral. Pertaining to the front or inward surface of an organ in relation to the axis, as in the upper surface of a leaf; adaxial. Figure 1428.




Therefore, dorsal/ventral are determined relative to the axis of where the leaf connects to the stem (ie., the axil). Because of this, in my experience terms such as abaxial and adaxial would be preferred when referring to plants.