Possibly one of the most annoying things is Romans speaking with an English accent. Romans spoke Latin, and quite unsurprisingly, Latin and Italian have a whoooole bunch in common. So why do they not have, er, Italian accents? Actually, "Rome' encompassed people from dozens of ethnic and language groups. It would sound more like modern New York-accents and languages from all over the Empire.
I recently read a published novel of WW2 submarine warfare where the author has his sailors saying things like "on your six" and " Gangway, working Navy here", both modern day terms.
Another where a Roman swung into his saddle by using the stirrup. The stirrup didn't make it to Europe until around 700 CE.
The seeming desire to assign cartoon villain status to Nazis and Confederates (even, I kid you not, as ghosts) rather than write them as living, breathing characters.
The tendency to assign modern attitudes and outlooks to historical characters. (I think there's a good deal of Mary-Sue ism here for a lot of writers. That's they way they think, so therefore their character should, too.)