I have heard Kyrgyzstan pronounced by Americans about every way imaginable, so here's a quick lesson on how to pronounce it correctly.
First you have to decide if you want to be totally authentic or not. The Russian pronunciation is close, but not quite right. However, it is easier for us to use the Russian translation:
keer-GEEZ-i-stan (the -i- syllable isn't quite there, but it isn't not there either).
For the Kyrgyz pronounciation, you use a tricky little sound that a bit like "uy" or maybe a cross between "oy" and "i." You've almost got to hear it to use it. It's a shorter sounding vowel than "ee." Also, the first syllable is stressed- KYR-gyz-stan. Those y's are where you use the other vowel sound. And make sure you roll that "r." And that "g" is a "g," not a "j."
The Russian pronunciation is fine, especially in comparison to the shudderingly awful "KIR-jiz-stan" that many Americans seem to like.