Ahh, cultural training. While I have not received the military version of cultural training, I have had a number of courses that emphasized different cultures, the elimination of "deficit thinking" and use of "community stores of knowledge", as well as the interaction between the various cultures.
Deficit thinking means that one should not focus on what the groups do not know, but rather on what they do know. Likewise, the community stores of knowledge. Thus, one should not emphasize that they do not know how to use a computer, but rather should note their ability to use the bus system to get to their various destinations. No, I am not making that up. Yes, it is a specific example from a specific class. But I digress....
None of this had any particular effect on attitudes or opinions. It did not improve harmony between groups or individual elements within those groups.
Always supposing, of course, that my comments weren't the problem.... 
My observation - knowledge of another culture does enhance the ability to get along with them, if one cranks up their SA and really works at accomodating the other viewpoint. However, such knowledge also informs one how to annoy members of other groups in subtle and unpleasant ways. The knowledge can form the basis for gaining useful cooperation from members of those other groups; however, there is a price to be paid for such cooperation. I do not doubt for a moment that Special Forces soldiers know how to turn that to good advantage. I am not at all sure that I, or most others, could.
All MOO, YMMV, etc.