In my own devotional life and study, I follow both the traditional Roman liturgical calendar and the Byzantine calendar, specifically that published by the US Melkites, whom I regard as the 'other' traditional Catholics (more on that later). It is a huge learning experience for me to compare the two calendars, as the arrangement of feasts reflects theological similarities and differences between East and West, illustrating the complexities of the faith and the richess of Catholicism.
Case in point: today, December 8th, we joyfully celebrate the Feast of the Immaculate Conception. My Byzantine Melkite calendar, however, does not have the feast. What it does have is the "Feast of the Conception of the Theotokos" or the "Feast of the Maternity of St. Anne", tomorrow, December 9th. Moreover, the feast of the Conception of the Theotokos does not celebrate her holiness or her immaculate nature, but simply that, her Conception. (The Byzantine feast that truly zeros in on her holiness is the Feast of the Entrance of the Theotokos into the Temple when she was a child, which is celebrated on November 21.)
I did some research and apparently the East had this feast first. It seems to have originated in Palestine in the 5th Century, and started making its way into the West in the 8th Century, first in the liturgical traditions in northern Europe (Sarum, Gallican, etc.) and gradually adopted down south by the Roman liturgy. At some point it was moved back one day to December 8th in the West. Originally the Western feast was called the "Feast of the Conception of the Blessed Virgin Mary," and much later the name was modified to the "Feast of the Immaculate Conception." The feast on our calendar, on December 8th, is exactly 9 months before the Feast of the Nativity of the BVM, which falls on September 8th. This is quite fitting, and actually mirrors the same reality in the case of the Conception of Our Lord and His Nativity, which are 9 months apart, on March 25 and December 25, respectively.
But the East does it slightly differently. Whereas we celebrate two conceptions, Our Lord's and Our Lady's, they celebrate three, adding that of St. John the Baptist to the mix. Moreover, the Conception of the Theotokos in the Byzantine calendar is on December 9th, so that it is not quite exactly 9 months prior to the feast of the Nativity of the Theotokos on September 8th, signifying by that fact that her conception, though a wondrous event in the history of salvation, was not perfect, not virginal. Many of the icons that commemorate the event represent this fact by showing Sts. Joachim and Ann embracing with a bed in the background (see the icon above). The same is true of the Byzantine feast of the Conception of the Forerunner, St. John the Baptist on September 23, which is 9 months minus one day before June 24 (the Nativity of St. John), and which is also represented in iconography by showing Sts. Elizabeth and Zacharias embracing.
Now, whereas the Conception of the Theotokos and the Conception of St. John the Baptist are both 9 months minus a day away from the corresponding nativities, in the Byzantine calendar the Conception of Our Lord, celebrated on March 25, is exactly 9 months before the Feast of His Nativity, December 25. This is to symbolize the perfect, divine, and virginal nature of His Conception, given that He was born not of the will of the flesh, nor of the will of man, but of God.
Now, all Easterners, Catholic and Orthodox alike, firmly believe, profess, and celebrate the fullness of grace of the Theotokos, whom they exuberantly call in the litanies of their Divine Liturgy "our all-holy, immaculate, most blessed and glorified Lady, the Theotokos and Ever-Virgin Mary": Τῆς Παναγίας, ἀχράντου, ὑπερευλογημένης, ἐνδόξου, δεσποίνης ἡμῶν Θεοτόκου καὶ ἀειπαρθένου Μαρίας (I've memorized this Greek prayer as an ejaculatory prayer, as I just love how the words roll off the tongue...). Also, in every Divine Liturgy they sing the wonderful hymn Άξιον Εστίν, which praises the Mother of God beyond all creatures for her excelling holiness:
It is truly meet to bless you, O Theotokos, who are ever blessed and all blameless, and the Mother of our God. More honorable than the Cherubim, and more glorious beyond compare than the Seraphim, you who without stain didst bear God the Word, you are truly Theotokos! We magnify you!
In other words, there is no doubt in the Eastern mind that the Theotokos was always completely free from sin and absolutely full of grace from the moment of her conception. This is Sacred Tradition and every corner of the universal Church is aware of that. Even Muslims have this as part of their tradition! But what Eastern Christians struggle with is the very idea of original sin, which although we believe it has been divinely revealed (Ecce enim in iniquitatibus conceptus sum: et in peccatis concepit me mater mea, Ps. 5o:7), the development of this dogma occured in the West without the participation of the East, and its subtleties have simply escaped them.
As far as I can understand, the tendency among Orthodox theologians is to think that what is inherited from Adam and Eve are the effects of original sin: we inherit from Adam our concupiscence, our passibility, our mortality. But it would be absurd, they think, to claim that the sin itself is inherited without personal consent. In other words, they don't conceive of any other sin than actual sin, and they can't conceive of a 'stain' of sin that is inherited without personal consent. Our nature is definitely fractured thanks to Adam's personal sin, but in their mind Adam's guilt is not our guilt. As a consequence, most Orthodox theologians flat out reject the dogma of the Immaculate Conception. That's one way in which some Orthodox undersand the matter. But others reject the dogma for other reasons, many of which are not always coherent, nor do they even seem to have a sufficiently adequate grasp of the dogma that they criticize. And above all they resent the West and its imperialist understanding of the papacy, so many will reject a priori any dogmas issued since the Great Schism, even if they have some equivalent doctrine within their tradition.
Byzantine Catholics, however, do not reject the dogma. Ultimately they accept it (otherwise they wouldn't be Catholic), but they assess this extraordinary act of Pope Pius IX's magisterium in different ways. Slavic Byzantines seem to have little problem with it, and some even celebrate the feast on December 8. But the Greeks/Melkites, who are arguably the more staunchly Byzantine of the bunch, the 'purists' as it were, seem to have some reservations regarding the dogma's implications regarding original sin, as they are strongly attached to their Eastern way of thinking. Not that they flat out deny the dogma: they are a lot more careful than that. They admit the truth of the dogma in its own terms, but prefer their own tradition and theological language, and many further think that it was unnecessary, imprudent, or inconsiderate for Pius IX to define the dogma in the way he did, lacking sensitivity to Eastern ways of speaking and thinking. Ultimately, as in other issues, they are walking a thin line between being fully faithful to their own tradition (which I as a traditional Catholic can certainly appreciate!) and being faithful to the pope, whom they profess to have primacy over the universal Church, despite the fact that he has often made decisions that have been detrimental to authentic Eastern modes of thinking. This thin line that they are walking is remarkably similar to the line that many traditional Roman Catholics are walking between being faithful to tradition and being faithful to the pope, the only difference being that the Melkites have been walking it for much longer than we have, since the popes whom they profess fidelity to have been undermining their Eastern tradition for much longer.
Melkites have the courage to continue preserving their own tradition. They will teach, profess, and celebrate Our Lady's perfection in grace, the "all-holy, immaculate, most blessed and glorified Lady, the Theotokos and Ever-Virgin Mary," in their own language and according to their tradition on November 21; and they will commemorate her conception on December 9, without reference to the Western understanding of original sin. It's the only way they can be traditional Catholics.