The records of Roman Catholic parishes might contain the following information about your immigrant Lithuanian ancestor (or his/ her close relatives) that would help you to find the link to your immigrant Lithuanian ancestor's birth family in Lithuania:
[My descriptions here are limited to records of Roman Catholic parishes, because (1) that is the extent of my experience, and (2) this website is primarily focused on finding ancestral records in the Roman Catholic parish registers in Lithuania.]
Of course, these parish records are also sources of "family tree" information about births, marriages and deaths in the US family of your immigrant Lithuanian ancestor and his/ her US relatives. The baptism records from the early years after your ancestor's immigration may identify children for whom a birth certificate was never issued (because it wasn't yet customary or required to have a birth certificate for a child born at home).
Other sacramental registers, such as for confirmation and first communion, might be of interest to family members but are unlikely to be of help in finding the link to your ancestor's birth family in Lithuania.
[For parishes other than the Archdiocese of Chicago parishes described on this website, members of the LGGS Yahoo Group on Lithuanian Genealogy (Reference 5) might be able to help you in searching for parish records.]
In general, these records are in Latin, but it is not difficult to translate the key information once you have learned certain patterns about the typical formats of entries in the parish registers . For a further discussion of this topic that includes examples of register pages and translated entries, go to my page about Chicago Lithuanian RC Parish Registers Online.
Marriage Registers. At the time of the US marriage of your immigrant Lithuanian ancestor, the marriage registers of the Roman Catholic parishes asked for the names (and sometimes the birthplaces) of the parents of the bride and groom. In general, the marriage registers were on pre-printed forms that were fairly standardized. The typical entry for the father's name is limited to the given name. The groom's father's surname is assumed to be the same as the surname of the groom. The bride's father's surname is assumed to be derivable from the bride's maiden name (refer to the description on this website About Forms of Lithuanian Surnames). The typical entry for the mother's surname is the mother's maiden name; however, sometimes the surname is not entered. Some of the marriage registers also asked for the birth places of the parents; if the birthplace is listed, it is typically a general entry like "Lithuania".
Death Registers. The death (funeral) registers of the parish often asked for the names of the parents of the deceased, but it seems that different priests had different ways of entering a response. It seems that the priest always entered the parents' names if the deceased person was not yet an adult. For deceased adults, the parents' names are often not entered but sometimes they are entered.
Marriage Registers. In general, the marriage registers were on pre-printed forms that were fairly standardized; almost all asked where the bride and groom were from, and some asked where the bride and groom were born. For those that asked where they were from, most priests would typically enter the name of the local neighborhood or town where the bride/ groom was residing just prior to the marriage, and sometimes they would instead enter a general location such as "Lithuania". Some priests, however, would enter the name of the home town or village in Lithuania (although not specifically stated as such, that would typically equate to being the "baptism town" or "birth village"). Similarly, for those registers that asked where the bride and groom were born, most priests would typically enter a general location such as "Lithuania", but some priests would typically enter the name of the birth village or its nearest town (which would be the "baptism town").
Death Registers. The death (funeral) registers of the parish often asked where the deceased person was from, but it seems that different priests had different ways of entering a response. Some left this blank, and some would enter the name of the local neighborhood or town where the deceased person had been residing in the time prior to the death, and some would enter a general location such as "Lithuania". It is possible that a specific birth place in Lithuania could be entered, but I haven't seen one.
General (re: "pre-Americanized" name). In general, finding a "pre-Americanized" surname depends on finding a record of your immigrant Lithuanian ancestor that is dated prior to the time that your ancestor made the name change (and you also have to have a way of knowing that is in fact a record of your ancestor). The pre-Americanized name will show up in both civil and church records prior to that time, and the Americanized version will show up after that time. In that sense, only parish records dated before the name change will show the name change. One important exception to this occurs if your immigrant ancestor was in an ethnic Lithuanian parish; the records of those parishes tend to have the pre-Americanized Lithuanian name for a longer period after the name was Americanized.
Baptism Registers. If you know where your ancestor's child was baptized and the approximate baptism date, you can locate that child's baptism record. [Most children of that culture were baptized within a few weeks after birth.] All baptism registers had places to enter the names of the parents, including the mother's maiden name. The names of the godparents were also listed, and godparents were often close relatives. (So you might also find your ancestor's name as the godparent of a child of a close relative.)
Marriage Registers. If you know where your ancestor was married and the marriage date, you can locate your ancestor's marriage record. The record contains the names of bride, groom, and at least 2 witnesses. Today, the names entered would typically be the names of the "best man" and the "maid of honor", but in records of the early 1900's both witnesses were often males. [I would guess that they were usually the "best man" plus the bride's closest male relative who was a member of the wedding party.] (So you might also find your ancestor's name as a witness at the marriage of a close relative.)
Death Registers. The death (funeral) registers of the parish typically asked for the names of the deceased person and the parents of the deceased person. If a child of your ancestor died, the name of your ancestor will be listed in the parish death register.
Some dioceses allowed the formation of ethnic (nationality-based) parishes. Immediately after immigration, all four of my immigrant Lithuanian grandparents became members of Lithuanian RC parishes in Chicago. There were also Lithuanian RC parishes in other cities. At that time in Chicago, it was typical for a Roman Catholic immigrant of any nationality to belong to a parish composed of persons of his/ her own nationality. Although at that time the Mass was recited in Latin, the sermons, announcements, various prayers, and other parish services and events were in the vernacular language. In the years immediately following immigration, the people in such parishes were becoming accustomed to using the English language, and some were "Americanizing" their names, but at parish services and events they were likely to be conversing in their native language. For this reason, these records can be a source of finding their "pre-Americanized" names that can be used to find the link to their birth families in Europe.
In general, unless the records are online (such as the pre-1926 Chicago records), you need to first find out where the records are kept and then send a request along with a donation to compensate for the time spent in fulfilling your request. There is no guarantee that your request will be fulfilled, even if your ancestor's records are there. In general, these older records are (1) still at the parish and/or (2) in the archives of the diocese.
If you don't already know the name and location of the parish of your ancestor, you have to figure it out based on where the ancestor was living at the time. You would first try to find the closest ethnic Lithuanian parish. If that was too far, you might try looking for the closest ethnic Polish parish. If that was too far, then probably simply the closest non-ethnic parish. The civil marriage license will contain the signature of the priest who officiated at the marriage ceremony, but won't necessarily identify the church.