Does anybody have a recommendation for a clean, small rental apartment in Kiev ? Does not have to be in the center, as long as it is close to a subway station.

You may try www.rentkievapartments.com. they used to have positive feedbacks.

Igorek, you probably already found apartment or maybe even visited Kiev.
Link to the information I posted on tripadvisor after my trip last May: http://www.tripadvisor.com/ShowTopic...Kiev-Kiev.html

For future trips, you might try Olga Apartments. Nataliya will work hard to find what you are looking for.
http://www.olga-apartments.kiev.ua/en/

I am not advertising for Olga Apartments. I used them last time I was in Kiev (August) and was very happy with them. I found it difficult on my previous trip to find a good apartment without getting ripped off and was very relieved to have the good service. But, use whatever company fits your needs.