I have never heard of that version being the 'common name' for the country. 'Czechoslovakia' yes, 'Czech Republic', yes; 'Czechia' or 'Bohemia-Moravia', no.
The Czech government recently formally introduced a short-form name of their country in English, "Czechia", and ran a little PR campaign to get people to use it instead of "the Czech Republic" all the time.
EDIT: as a side comment, the country's name in their own language has been "Česko" for a long time (compare "Česká republika"). Wiki also claims that Czechia has been around for a while, though never popular.
As an aside, it's interesting to note that "Czech" uses "Cz" instead of "Č" because the former is an archaic spelling and the name was brought into English before the spelling reforms that replaced it with the latter.