Грамматика русского языка. Часть 5. Особенности произношения. Правила. Russian Grammar. Part 5. Special Pronunciation Rules.

Even people who have studied Russian for many years frequently don't pay enough attention to some of the finer points of Russian pronunciation. Therefore, even it you're a more advanced student, study the following rules carefully.

The First Rule

After the letters ж, ш, and ц, the letters и, е, and ё will sound like ы [ih], э [eh] and о [aw].

жить [zhiht'] to live
жест [zhest] gesture
жёлтый [zhawl-tihy] yellow (m.)

шить [shiht'] to sew
шесть [shehst'] six
шёлк [shawlk] silk

цент [tsehnt] cent
цирк [tsihrk] circus
цифра [tsihf-ruh] number

The Second Rule

After the letters ч and щ (which are always pronounced soft), а, о, and у will sound like я, ё, and ю (minus the initial й sound).

чай [chahy] tea
час [chahs] hour
чулок [choo-lawk] stocking

щука [shchoo-kuh] pike
чаща [chah-shchuh] thicket
ощущать [uh-shchoo-shchaht'] to feel, to sense

The Fird Rule

The letter г sometimes sounds like a в.

его [yee-vaw] his
сегодня [s'ee-vawd-n'ee] today

The Fourth Rule

The letter ч sometimes sounds like a ш.

что [shtaw] what
конечно [kuh-n'ehsh-nuh] of course

The Fifth Rule

Sometimes, in a group of three or more consonants, one of them is not pronounced.

Examples:

солнце [sawn-tseh] sun, not [sawLn-tseh]
честный [chehs-nihy] honest, not [chehsT-nihy]
сердце [s'ehr-tseh] heart, not [s'ehrD-tseh]
известный [eez-v'ehs-nihy] known, not [eez-v'ehsTnihy]
чувство [choos-tvuh] feeling, not [chooVs-tvuh]
здравствуй [zdrah-stvooy] how do you do, not [zdrahV-stvooy]

CONGRATULATIONS! 

Now you have learnt the fundamental principles of Russian pronunciation. If you continue to practice and apply these rules consistently, you will pronounce Russian correctly.