It depends if you are Roman Catholic or Eastern Catholic. In the Code of Canon Law which is for Roman/Latin Catholics, it explicitly states that you need to attend a “Mass” of any Catholic Rite (Eastern Rites) on a Sunday to fulfill your obligation. Note that if it is not possible for you to go, then there is no obligation. So if you are in a small town in Russia and there is only an Orthodox parish there and there is no Catholic parish in any direction for hudreds of miles, then you have no obligation to go to an Orthodox Liturgy. You can and it is to your spiritual advantage to do so, but you are not obligated in any way.
In the Eastern Code, that provision is absent. It says that one must attend Divine Praises (at least one of Vespers, Matins or Divine Liturgy), but there is no clause of “any Catholic Rite”. It is up to each Eastern Church to state those provisions, and the reason it is absent is that in areas of great persecution, like the Middle East, there is actually intercommunion in some places (like between the Chaldean Catholics and the Assyrian Church of the East) so you can effectively go to their Liturgy for your obligation. But this is in specific locations as mandated by the respective sui juris Churches.
The other way to go about this is if you have Orthodox family, you may want to just ask for dispensation from your pastor if you are going to Divine Liturgy with them at certain times of the year (like Christmas or Easter).