Speaking as a genealogist, changes like "Kato-->Kate" are actually much more common than direct translation (at least in terms of historical immigration patterns, these days most people keep their surnames).
The most common thing was to simply change the spelling/pronunciation of the name (long/difficult names often get shortened). I know someone whose family moved from Italy to Russia several generations back - his surname sounds Russian to my untrained ear but it's actually a Russianised version of an Italian surname.
Another friend goes by a shortened version of his first name because the full is hard for English speakers to say correctly (in comparison, he gave his son a quite British sounding name although apparently it's one of those works-in-both-countries cases).
The alternative is to simply lose the old name entirely and pick a new one. If you've married into a local family, taking your wife's name is always an option (Lafcadio Hearn, anyone?). For first names, just pick one to use - when I was in high school I thought it ever so cool that my Chinese friends had gotten to pick their own names.
I think there is a reason for this - most people don't think about the meanings of names or think of names in terms of their meanings very often. If you said "Nakamura" most people would think immediately "a person's name" or maybe of a particular Nakamura they know, they wouldn't think of the meaning.
Similarly "Middleton" = a person's name (or "when are they going to stop covering this damn wedding on every channel?"), not "middle town".
(Technically, isn't her surname now Windsor? The meaning that, I believe, is "quick, distract the populace from the fact that the British royal family is actually rather German")