Can US citizen have dual citizenship with Czech Republic?

The short answer appears to be no, as the Czech Republic asks for a renunciation of your previous citizenship when you are applying for Czech citizenship. There are some exceptions to this:

- the country of which you are currently a citizen will not let you renounce your citizenship

- the country of which you are a citizen does not issue certificates affirming/asserting renunciation (so you cannot provide proof of the renunciation)

- renouncing your current citizenship would put you in physical danger

As for the other way around, if you are a Czech citizen and you are voluntarily naturalized, you forfeit your citizenship. If, however, you acquire citizenship through being born in a country, or through marriage, it seems as though you could actually be a dual citizen.

There are supposedly numerous people who have not notified the Czech authorities of their new status as citizens in other countries and have retained their Czech passport, so they are (at least for a time) de factor if not de jure dual citizens.