Former Protestant- we always ended our prayers with "In Jesus' name we pray, Amen." or just "In Jesus' name, Amen."
I remember a radio preacher once emphasizing that Christians are only to pray to the Father, and that Christians merely pray in Jesus' name (or ask things in his name, like the OP) who serves as a mediator. Most Protestants don't go that far in saying prayer is only to the Father and not the Son, but I think the reasoning is similar in affirming Jesus' role as mediator between us and the Father.
Which leads me to realize that much of my prayer experience growing up was oddly subordinationist or binitarian, with its distinct lack of the Holy Spirit or just asking to acquire it as if she's some sort of energy.
Anyway, to the OP - I'm not sure about OO, but I haven't seen it among EO that I can recall. I have, however, seen it among some Latin Catholics.