This question occurred to me after reading this question about the observance of Lent among Eastern Rite converts.
It is my understanding that since 1582, the Roman Catholic Church has followed the Gregorian calendar.
The calendar having been a product of the post-Schism Roman Church, Eastern Orthodox were loathe to accept it - some for 5 centuries. Some majority Eastern Orthodox countries (e.g. Greece) were still subscribing to the Julian Calendar that had been in place during New Testament times as late as 1923.
While all Eastern Orthodox countries (as far as I know) today observe the Gregorian calendar for civil purposes, some Orthodox jurisdictions (e.g. Russia) still observe the Julian calendar for religious purposes. Within the Eastern Orthodox Church we have the somewhat bizarre situation of virtually all (if not all) jurisdictions observing the Julian calendar with respect to the Triodion and the Pentecostarion - the periods leading up to and including Easter ("Pascha") and Pentecost, respectively - but some holding to the Gregorian calendar otherwise.
Given the affinity between Byzantine Rite Catholics and the Eastern Orthodox Church, I was wondering if there are any Byzantine Rite Catholics or other Rites that at any time during the year observer the old Julian calendar for some purpose; perhaps along the line of New Calendar Orthodox jurisdictions more or less holding to the Old Calendar during the Paschal season.