I take it you're looking to expand beyond the Byzantine/EO churches, then? I know that they use the liturgy of St. James on his feast day, at least (I think some other days, too, but I can't remember which), but it is not their primary liturgy. It is the primary liturgy of the Syriac Orthodox and the Orthodox in India (also Syriac by rite and history, though their native language is largely Malayalam, one of the Dravidian languages). So your best bet is to look to them. Zmirotho
has a small page of Syriac liturgies available to listen to or download, though they seem abbreviated to me (the full liturgy of St. James is very long). You will notice that they are not completely in Syriac, either. This is not unusual.
For Coptic liturgies (St. Basil, St. Cyril, St. Gregory, of which St. Basil is the common one), Tasbeha
is your best bet. The Ethiopians once had as many as 14 anaphoras received from Egypt, but over time these were reduced to the three in common with Egypt (Daoud, 1959). Interestingly, the Copts of Egypt until fairly recently also had a Coptic/Arabic translation of the Ethiopian liturgy that was performed. I don't know how often, but it was recorded and you can listen to it on Youtube prayed by Abouna Estephanos Rizk
. I'm definitely biased, but I think it's amazing. As it was explained to me, it is no longer performed because HH has asked that Copts stick to their own forms of liturgy.
That's about it unless you're interested in the anaphoras associated with the Nestorians and those of that liturgical patrimony (the Malabar and Chaldean Catholics), who as far as I understand use most often the liturgy attributed to Addai and Mari, the apostles to the East Syrians in the Persian empire. I don't own any recordings of this (they're tough to come by), but you can watch broadcasts from the Chaldean cathredral in El Cajon, CA (St. Peter's, if my memory serves me) via Kaldu TV
on the weekends. They stream their liturgies live, as well as cultural programs throughout the week. I used to watch the liturgy on a semi-regular basis (my old FoC in Oregon is friends with the priest who serves there, Fr. Bazzi), but since I stopped when I moved to NM I no longer remember when they're broadcasting the liturgy. Check the schedule on their website; hopefully it is accurate.
Edit: I didn't mean to make it seem like I'm overlooking the Armenians. I didn't comment on them because I don't know much about their church and there are so many albums of Armenian liturgy that there's really no way to know where to start unless you happen to have a favorite. I personally love the compositions of Komitas/Gomidas Vardapet, who seems like he's probably the most famous of the old names connected to the Armenian church music. Some of his material has been reissued on CD by the Traditional Crossroads label. Other than a few 78s of him, the only Armenian liturgical music I've had the pleasure of hearing outside of Youtube is a double album c.1961 called "Christmas in Bethlehem and the Deacon's Litanies in the Armenian Liturgy". I've seen a few copies for sale on E-bay for $10-$15, if you happen to have a record player. It seems authentic to me, in that the entire first record was actually recorded in Jerusalem (the second has some Vardapet pieces recorded at a church in New Jersey, for some reason), and sounds every bit of its 50-60 years of age (in a good way). But there is probably more commercially-available material from the Armenian Orthodox Church than any other non-Byzantine church out there, so you'll have more than you ever bargained for with a 5-10 second Amazon/Google search.