There are lots of answers to this question already, but I felt rather strongly about this question and have created an account on this Stack site just to provide my own opinion on it.
You never actually state how big your workplace actually is.
I work for a global telecommunications company, which operates out of 170 countries. My manager is on the other side of the world to me (I'm in Northern Ireland, he's in India). When I email him, I frequently don't get responses.
But here's the thing. He is receiving hundreds of emails an hour. He has over 15,000 unread emails in his Inbox. It's not a simple case of saying he's not doing his job properly. Given the size of the company, his seniority within the company and the complexity of the role he has, he is simply inundated with information on a constant basis. Whilst it can be a frustrating situation, it's the nature of the beast with such a large company.
Therefore, this is how I handle it. If I want time off for something, I send him a meeting request for a week, simply stating that's my holiday time. As his Calendar, like many senior managers, dictates his life, he will notice it. For almost any other urgent enquiry, I'll ping him on MS Lync or call him. But all other actions which aren't urgent I can either wait for a response and if I don't receive one, I can casually bring it up on our weekly/biweekly team knowledge calls (which should be taking place if you're that far apart, just to touch in with what has been happening).
If I'm completely stuck with an issue and he hasn't responded by email, I don't get frustrated and wonder what to do next. I escalate it! I wouldn't go straight to HR to complain or fret about how I've worded my email - not because those things aren't important, but because I could be making an absolute mountain out of a molehill.
I'll simply try a different approach to get in touch with my manager (i.e. Lync/calling), discuss what I want to discuss and find out if there's a different form of communication he prefers or would be quicker to respond to. I always find this is the best way to approach a situation rather than panicking unnecessarily first.
If you still can't get in touch, try and figure out why. Is he deliberately avoiding you, in which case I would say go to HR. Is he too busy? If so, what are the standard policies in the workplace for escalation? Find them out and follow them. Also bear in mind the size of the company and your manager's workload. In my case, it's unreasonable to expect a response to anything but an absolute priority (i.e. a major system failure) within a few hours. So if I want to email him about, for example, time off - I wouldn't do it the day before I wanted it. It would be completely unreasonable to expect a response in that timeframe.
The key thing in any workplace is to know the rules and simply follow them. If you do that, you're doing nothing wrong. Rest assured in that fact.
One final point:
I insist on straightforward communication and I refuse to take hints
so I act as though I don't get them.
If this is your attitude, confirm any communication you have. If you feel you are being hinted something, simply repeat back what you think you're being asked to do and ask for confirmation.