The usual badge or cognizance of the York family in the early years was the Fetterlock, usually with a falcon inside it. By the way, if anyone wonders what a fetterlock is, there's a nice picture here.
There were no white roses anywhere. This device seems to come from the Mortimer family and is traditionally associated with Clifford Castle which is near Hay-on-Wye. The first member of the York family qualified to use it would be Richard, the third duke.
The first two dukes also made some use of the ostrich feathers, more usually associated with the Black Prince, but used in various ways by all Edward III's sons.
It is said that Edmund of Langley chose the closed fetterlock as a badge because he felt that he and his family were shut out from the throne by senior claims. Be that as it may, it is a fact that after the accession of Edward IV, the fetterlock was always shown as open.