Just act like you would in any large city and be street aware. I wouldn't be surprised if you got some stares, because blacks are still rare in Russia, although the stares will probably come not from Muscovites but from visitors to Moscow.

Keep your passport on you at all times. I've heard accounts of cops harassing blacks from African countries studying here. I think an American passport will be enough of a deterrent for cops to not give you problems.

Also, avoid skinheads.... cops don't deal with them either.

I think you will be safe if you use your streetsmarts. Let us know how it goes.