Another random pronunciation tip: Try to relax your tongue and lips when saying Russian consonants. English speakers strain their mouths when saying consonants.

For instance, try saying these English words:
kit
pot
tap

Say them while holding your hand in front of your mouth to feel the air that comes out. When pronouncing the letters t,k,p, you'll feel a puff of air come out. It's called aspiration, and Russians don't do it with these consonants. Try practicing saying these words to where you don't feel the puff of air (you're going to feel air, of course, because you're breathing out when you speak--but you shouldn't feel a sharp puff of air).

(These consonants are usually only aspirated at the beginning of a word. For instance, feel the difference between the t in 'tap' in and the t's in 'butter')