In my ‘vast' experience, there are two keys to directing Shakespeare for a modern audience:
1) Physicalize the transitions
2) Everything works on stage
By physicalize the transitions, I mean identify the emotional and intellectual transitions both within and between the lines, and use a physical change (blocking, business, set, lighting) to emphasize the change. This is the only compromise to you need to make to help the audience comprehend the story. You don't have to set it in Nazi Germany or the Stone Age or the Far Future, all of which is usually a distraction after the first five minutes anyway. (If anyone can think of a better word for physicalize, I'd be grateful.)
By everything works on stage, I mean assume that whatever is written will connect with the audience if properly performed. Shakespeare wasn't having a bad day or writing for a different era or trying to solve a theatrical problem. If it's in the play, it works on stage, then and now. If it isn't working, it's your problem, not his.