Q: Which is correct (if any): "please let me know what do you think"? or "please let me know what you think"? I find myself unsure which of the following is more correct:

please let me know what do you think

Or 

please let me know what you think

My gut feeling tells me that it's the latter...
Or are both incorrect and there is a better way to say the same thing?

A: The latter is correct and the former is definitely wrong. 
When you construct questions in English, you follow this pattern: V + S, that's why you have the Helping Verb "Do" there. 
But when you construct sentences, it's the opposite, you have to follow: S + V. And if you add any other Noun Clauses or Noun Phrases to your sentence, whether they were originally in question form, they cannot follow the "V + S" question pattern. 
Take a look: 
What are you thinking? = Question 
Please tell me what YOU ARE thinking. = Sentence
Notice that I switched the Verb and the Subject in the original question. 
In the same way: 
What do you like? = Question
Please tell me what you LIKE. = Sentence 
Where did you go? = Question 
Please tell me where you WENT. 


*

*"Do" and "Did" as Helping Verbs are normally used only in Questions and Negative sentences, not positive sentences


This topic also encompasses Direct and Indirect Questions 
http://www.coolelf.net/2011/12/are-you-direct-or-indirect.html

A: 
Please let me know what you think

is correct. Even if do were to be included, the correct form would be

Please let me know what you do think

but this should only be used for emphasis (for example, to contrast with don't think).
