Schwurgericht n ≈ criminal court with 3 professional judges and 2 lay judges dealing with serious crimes (at Landgericht)
There are recommendations by the Auswärtiges Amt (Federal Foreign Office) online for translations of court names such as Landgericht (see the list of PDFs at Terminologie). People might want to use these. But there are no recommendations on translations of the names of the various types of court or chamber within the courts.
If you translate Landgericht as Regional Court, as the FFO suggests, it seems a good idea to me because it does not sound like any existing UK or US court. The terms local - regional - higher regional suggest a hierarchy too. The original name Landgericht should be retained somewhere in the translation though, for clarity.
So what about Große Strafkammer? That's a court (chamber if you like) that deals with serious criminal matters and it sits with three professional judges and three lay judges. (Sits with makes me think of the joke: Mit ihm ist ein großartiger Mensch gestorben - Was, zwei Menschen in einem Grab?). You could say Large (?) Criminal Court/Chamber/Division. You might be able to avoid this detail depending on the context of the translation.
Schwurgericht is a kind of Große Strafkammer which deals the very most serious offences. It has no jury. Is it worth keeping the German word and defining it?
At all events, it's a superb example of useless translation advice on the Web.
A discussion on LEO rejects jury court but contains suggestions of magistrates' court (a lower level and a specific UK term) and trial court (US term for a court of first instance - you might use it in some contexts but it's rather broad).
A discussion on ProZ has some good suggestions but the asker goes for a bad one. (I find those discussions useful sometimes). Crown court for a German court is mad (comes from Romain, who is usually good), and court of assizes is also too specifically UK, apart from having ceased to exist in the early 1970s. A non-member writes:
I came to a conclusion from all the opinions stated hereunder and that is:"trial by jury according to the German legal System". Do you agree?
Well, not really. I see I've given my two cents' worth in the past, to a poster who said there are twelve jurors in the Schwurgericht - this was apparently the case in 1879.
It does begin to look as if jury court was an appropriate translation in the late 19th century, and as if certain terms are enshrined in dictionaries and fester away there for centuries.