JKR explained in a web chat that Death Eaters couldn't produce Patronuses because they didn't need them.
Samantha: Was snape the only death eater who could produce a full patronus
J.K. Rowling: Yes, because a Patronus is used against things that the Death Eaters generally generate, or fight alongside. They would not need Patronuses.
Essentially, they're allied with the Dementors so they don't need a way to defend themselves from the effect they have on them, apparently. It's possible (though this is entirely speculation) that even if they were to attempt to learn, the experiences that led to them wanting to become Death Eaters might mean they don't have a sufficiently happy memory to make the spell work. Snape's extraordinary love for Lily is the exception; it seems like it's probably the only (very) bright spot in an otherwise miserable life.
All of that said, I'm not sure they would have ever been taught. Patronuses may not be part of the syllabus at Hogwarts, even at NEWT level. Harry learnt how to conjure a Patronus in his third year, and he taught a number of other students during sessions with Dumbledore's Army in his fifth year, but there's never a point where they actually learn the spell during a regular lesson.
They cover the theory of it and what it's used for, but there's never any practical sessions on casting the spell. There's also repeated mention of how extremely difficult it is to produce a corporeal Patronus, how impressive it is that Harry's able to conjure one, and how many (most?) qualified witches and wizards aren't capable of doing so. Seems counter productive to attempt to teach a spell that a large percentage of the students may never be able to perform adequately.