As far as I know, God hasn't yet handed out stone tablets containing nutrional advice to Abraham or any one else.

This is a ridiculous statement.

Believe me if God was to appear to me and tell me what to eat, I'd listen

The Word of Wisdom does indicate that flesh is ordained for the use of man, but to be used sparingly. It is also pleasing unto God that they should not be used, only in times of winter, or of cold, or famine.--D & C 89:12-13
While many of us have heard and read these scriptures for years, let's not stop at verse 13. Verse 14 says that all grain is for our use, "to be the staff of life." Verse 15 says "And these hath God made for the use of man only in times of famine and excess of hunger."
This verse does NOT refer back to verse 12.
Grain, whole grain IS good for man, again not to be used in excess and certainly not to be used all stripped down to the point that the nutritive value has been depleted to the point of a simple carbohydrate.
The same goes for fruits. When the fruit is taken, stripped of the natural fibers and turned into juice, YES, JUICE, the end result is fructose, which turns to fat in the digestive process. When fruits are eaten, use small portions and eat them as they come off the tree (or plant.)
A low or no carb diet can be very beneficial, especially to the LDS culture. Atkins may not be the answer, but low or no carb lifestyles can be utilized without the excessive meat. How about more vegetables and eggs instead?

The syntax in The Word of Wisdom is extremely poor (I wonder what fact-free English major Lisa Fraser would have to say about that), but I take this to mean that God wants us to eat flesh sparingly year 'round and ONLY eat grains when excessively hungry.

Well I'm not hungry on my usual 3 oz of flesh per day, so I don't eat or need to eat grains - to do so when not hungry would displease God! So God is OK with the Atkins diet.

Of course many LDS-followers eat grainy diets, are always hungry, so can eat even more grains without displeasing God. So the Ornish diet is OK with God too.