It's all about where the stress lies in the word.

восемь - vo-syem' - The "е" is unstressed here and sound like a "и" a bit.
девять - dye-vyat' - The "я" is unstressed and sound like "и" or sometimes "е" a bit.
десять - dye-syat' - Same as above

шестьдесят - shyest'-dee-syat - The "е" in шесть is unstressed and sounds like "йи" somewhat.
семьдесят - syem'-dee-syat - The "я" in десят is unstressed and sounds like "е" a bit.
восемьдесят - vo-syem-dee-syat - Same as above
девяносто - dye-vye-no-sta - "е" and "я" sound the same because they're unstressed.

Note that these unstressed vowels don't sound like the letter they are. They're usually softened or hardened depending on where it lies in the word. When pronouncing something, think of the where the stress lies and focus on that first. When speaking with natives, you'll hear numerous variants of the sounds because there's no emphasis put on them and everyone sound a bit different.