No, rule number one is try and look Russian. The good thing about traveling in Russia rather than say India is its not hard to blend in. (Unless your in the Caucasus)
I'm caucasian.
Well they got that wrong, cos you're obviously white.
That whole calling people as far away as Scandinavia Caucasian is odd as they have quite different characteristics. But the main reason why kiwi's would stand out in the Caucasus is that we don't tend to have that really thick black beard which is common in Islamic countries.
I've spent a good chunk of time in the Caucasus though, including being only a couple of hundred metres away from Georgia when the war was on which was a bit harrowing, and haven't had any problems. But it's one of the only places where you probably want to make sure they don't know your slavic, although the NZ embassy was pretty upset with us for going and said that if we were kidnapped by terrorists the NZ government wouldn't do anything to help.
Sochi is of course in the Caucasus as well, but it's a different boardgame. I haven't been to the city but obviously with the Winter Olympics, football, and F1 it's well policed and safe.
The other thing I thought of for people traveling to Russia is don't be afraid of the local cuisine, Russian cuisine has a bad reputation of being bland, and I don't like a lot of the formal stuff (caviar, aspic, that sort of thing) but a lot of the food is influenced by the extent of the old empire. Russian fast food is great. Shishlicks are like shish kebabs and are cooked in a special coal fire in vineger and wine. Kasha are porridge like things. Blini's are stuffed pancakes. Pelmeni are dumplings. You can get stuffed potatoes and european style hotdogs. Of course there are McDonalds, Hesburger, and KFC but the street food and fast food is great and healthy.
Also be prepared for a surprise if you eat old favorites, everything is modified to fit the Russian taste more than foods are modified for the Kiwi pallet here. I had Spaghetti Bolognaise from an Italian restaurant in St Petersberg and it didn't have any tomato in the sauce. I had Chinese literally 50 metres away from the border with China and it was pretty much Russian cuisine. If you want a mild curry in Moscow you ask for it to be extra hot. A good chain if you don't want to try a random place is my-my (pronounced moo moo) which is buffet style dining but very cheap and of a very high quality. In Russia you pay for food by the gram in restaurants rather than fixed price (even when they are fixed price the menu will tell you how many grams it is) so you pile up your plate, they weigh it, and you pay. Also a lot of places outside of the CBD don't have any signage, if your in the suburbs you kind of just have to watch what buildings people are walking in and out of and guess that they are shops, restaurants, etc. Also if a very attractive girl comes up to you in a bar starts flirting with you, before telling you you'd look really cool if you were smoking ignore her, she works for the cigarette companies.
If your traveling between the venues (I think it's just in Sochi, Moscow, and St Petersberg?) a good way to travel is to take the sleeper train. You leave from the centre of a city, wake up in the centre of another city and sleep on the way, the landscape in Russia is very flat and very large, and it's not unusual to stare out the window at unbroken forest for several hours. They have restaurant cars, bars, etc. Coupe is where you share a locked room with four people, Plaskart is like a backpackers with open bunks, people share food, booze, and stories and have a great time there. You get to know interesting people and learn the unique things to do. You can definitely take an overnight train from St Petersberg to Moscow, I don't know about Sochi. You can take an overnight train to the city of Rostov on Don or Mineral Waters and then catch a bus but I'm sure they put in lines down there for the Olympics.
Also Moscow is a crazy place, officially its around ten million people but unofficially it's above fifteen million and is by far the biggest city in Europe and is very wealthy, the average wage in Moscow is higher than Wellington or London. Because of the huge discrepency between the European part of Russia and the rest there are internal travel restrictions as they struggle to keep Siberia populated and a lot of Russians live and work in Moscow illegally. It has the best and worst of things in the world and huge excesses. Enjoy the old city hang out on Stary Abart, etc. But also go and checkout the nightlife - it's insane.