Have your Russian courses given much time to pronunciation?
In my experience, beginning Russian courses spend some time on the basics like hard vs. soft consonants, reduction of unstressed vowels, and voicing. After that, it's all we can do to manage case endings, aspect, verbs of motion, and more...
Yet developing a natural pronunciation does require more than what you get in those first weeks. As a 3rd-year student in St. Petersburg, and later as a graduate student at the Pushkin Institute in Moscow, I had the opportunity to take phonetics classes where we did finally have time to work on some important nuances.
I've just released a new pronunciation video on Youtube that touches on one of those nuances: how a consonant can influence the sound of a following vowel.
That may seem counterintuitive, since we're used to looking at vowel letters to tell whether a preceding consonant is hard or soft. But in terms of sounds, we see that consonants can indeed influence how a vowel is pronounced! And that explains why the letter и will sometimes be pronounced as if it were ы.
To keep things from getting too dry, I've included examples from poems by Хармс and Ахмадулина. You can find the video here.