One way for native Russian speakers to reduce their accent is to change the way they make the R sound. The Russian R is very distinctive because it is trilled or rolled as the tongue taps quickly on the gum ridge behind the upper teeth.
American English speakers do not tap their tongue on the gum ridge or on the roof of their mouth when they make the /r/ sound.
To articulate the American English /r/ pull your tongue back in your mouth with the tip of the tongue curled up. Your tongue should not touch the roof of your mouth.
Practice by saying: er, er, er. Notice how your tongue feels as it sits in the middle of your mouth.
Now say: run, read, farther, card, & lawyer. Remember that your tongue should not touch the other parts of your mouth when you make the American English /r/ sound.