If I understand the question, YES! Marriage (first marriage that is) does not constitute any impediment to ordination to the diaconate or priesthood in the Eastern Orthodox Church. Ordination to major orders does constitute an impediment of sorts to marriage, so a priest may not marry (after ordination, obviously) and remain a priest. Any candidate for ordination must meet a series of requirements, of course.
I cannot speak for the Orientals but I think the practice is the same.
As far as Eastern Catholics are concerned that is another story and varies depending on location.
This folder is for Oriental Orthodox discussion, note: in English the conventions may be confusing. Eastern Orthodox refers to the Byzantine Orthodox Churches while Oriental Orthodox refers to the non-Byzantine Churches. Note: there is not communion between them. Example: The Russian, Greek and Serbian Churches are Eastern Orthodox. The Armenian and Coptic Churches are Oriental Orthodox.