Firstly I don't want to sound like we are expecting to be confrunted by the KKK but I have heard some bad things and just want to set my fiancés mind at rest.
We are a mixed race couple I'm white british and she is Australian of Vietnamese desent. We are planning on visiting Moscow and Saint Petersberg.
We will mostly be in tourist areas and we are not ones who will be hitting any clubbing scene hard.
We both really want to visit and enjoy what looks like a beautiful country but feel it is only prudent to ask the question before we go.

Many people who live in Eastern Russia look quite Asian, and they are scattered across the former USSR.
I dont think you'll have a problem....
Problems may arise if you look like you come from the southern republics like Tajikistan, which Vietnamese dont.
The chances you meet the racism is a bit less if you meet the bear playing balalaika on the street )))
1st, as guys above have said, there is a lot of Asian-looks native people in Russia and Vietnameses look like almost similar. Next, there is a plenty native Vietnameses in most Russian cities having restaurant business here, and no tenses still occurred (just 'cause for the Russians their behavior is acceptable).
Regards - Igor

As a dark Belgian (passportholder), living in the Netherlands I have met less racism behaviour during many visits to Moscow than in the socalled tolerant Amsterdam.
Russia as a country ( and as a part of exUSSR) is very multinational and racism is something really out of average person's frame of mind.
I live in Buryatia (lake Baikal area) and this area is originally populated with Buryats which I am. We are Asians, and when I come to Western part of Russia there is absolutely no problem, even advantage sometimes (people are more interested in you).
Denis S.