Saturday evening, we attended the Liturgy at the local Melkite Catholic Church.  I’ve shared my observations about Eastern Catholic liturgies before, and have really nothing to add except:
1) Please go. If you’re a Latin Rite Catholic and have never experienced worship in one of these other Catholic churches, go.  It will open your eyes to what worship is, and help you grasp what people are saying, even in the Latin context of “singing the Mass” instead of “singing at Mass,” for almost every bit of the liturgy is chanted by presbyter, deacon and people, mostly facing in the same direction. It puts staring at each other and mumbling back and forth in a whole new light.
2) The Divine Liturgy is not directly comparable to the Roman Rite (either old or new) in many respects, but one of the points it might jostle about in the Westerner’s brain is that of the respective roles of clergy and people. One cannot accuse the Divine Liturgy of not being conducive to lay participation. (ritual caveat – yes, I know “participation” does not mean “vocal.”) It demands it, and it is constant. There is, to put it bluntly, never a dull moment. But at the same time, behind the iconostasis, there is a great deal going on. As we are chanting our litanies, the presbyter is praying, and at various points, he emerges with an invitation to us, and we pray something else, and then he returns to his work. I compare this to some of the Masses I attend in which, in the name of participation, I have to sit for at least five minutes after Communion is finished, in silence, watching the priest cleanse the vessels. Because “participation” apparently keepin my eyes glued on every act that the priest performs. Well, yes I’m supposed to be praying, but I’m going to admit to you, it’s difficult, when all is silent and there’s the priest, right up front and center, taking care of things.
3) That said, there is a casualness to the Eastern liturgy that is both refreshing and unnerving. The church is practically empty at the begining, but gradually fills up, until most people have shown up by the Gospel or so.  Because there is this (what I call) freight train aspect to the liturgy – you get on and it just goes – it’s hard for a Westerner, accustomed to associating “prayerful”  with silence and kneeling, to re-associate it with chanting and standing. But good.
(But I will say, too – that the other interesting thing here is that if you are tired of standing and you want to sit, it’s fine. There were times when about half the people were standing and the other sitting. It reminded me a bit of going to Mass in Italy. There is a sense that we are all here together, celebrating the feast, encountering the Lord, but we are still who we are,and that is fine.)
4) Very often, in apologetics discussions, you will read Protestants hostile to Catholic teaching on Mary hold up the Orthodox/Eastern view as preferable and something they’re more comfortable with. I have never understood this, and can only conclude that those who say such things have never actually experienced Eastern Christian spirituality. Certainly, there are points which are not dogmatized  – points which would be the center of the Orthodox argument against the Catholic approach –  but honestly now.
(edited to remove three uses of variations on the word “certain” in a single sentence.)
Often you get the impression that some people think that Marian devotion was invented by ignorant medieval peasants so distant from God and the Gospel that they turned to the primeval Mother Figure instead and just started mumbling random prayers to her out of the blue.
Well, no. Of course, Marian devotion began in the East mostly because, well, CHRISTIANITY began in the East. I’m not going to do some big historical survey here, but I’ll just say that any Protestant thinking that the Easterners will make them more comfortable in terms of Mary than the Romans, might want to take a look at this portion of the Liturgy of St. John Chrysostom, which is chanted directly after what we in the West would call the consecration:

PRIEST: Again, we offer You this spiritual worship for those resting in the faith, the forefathers, fathers, patriarchs, prophets, apostles, preachers, evangelists, martyrs, confessors, ascetics, and for every holy soul who has run the course of life in faith.   Especially, for our all-holy,. spotless, most highly blessed and glorious Lady the Mother of God and everlasting virgin Mary.
PEOPLE: It is truly meet to bless you O Theotokos who are ever blessed and all blameless and the Mother of our God.  More honored than the Cherubim and more glorious beyond compare than the Seraphim you who without stain did bear God the Word, you are truly Theotokos we magnify you.

In addition, the prayer, “Through the prayers of the Mother of God, O Savior, save us.” is a component on one of the repeated antiphons.
If anything, honoring and asking for Mary’s intercession is even more deeply integrated into normal Eastern liturgical life than it is in the West, because it is a part of the Sunday liturgy in a way that it is not in the West.
5) Do not promise your 4-year old that he will get bread at the end of the Liturgy unless you are very, very sure, they are not going to run out before you get up there.
(Photo above is from the end of the Liturgy. You go up to the front, kiss a cross held for you, then take a piece of what is called “antidoron” – bread that had been prepared for consecration but was not used.)