Is there a block above the note block?

Try hitting it with a redstone.

Is it possibly you crafted a jukebox by mistake?
They look very similar, though aren't used the same way.

When crafting it did you use a diamond in the recipe? If so, you crafted a jukebox, not a note block.

(Clarify?) You may have to interact with them first.

Hitting it with the right trigger will play the note while hitting it with the left trigger will change the pitch.

Have you tried making a noteblock in a different world just to see if that particular world is bugged?

Have you checked your settings to make sure all volume levels are turned up accordingly?

I had this same problem. I created some note blocks and they worked just fine, I even set them up to play the first phrase of the super mario bros theme. Then all of a sudden, they just stopped making noise, and when I hit them with the left or right trigger it would just break them, not play them (note: I saved my level many times after they stopped, so it is ok to save while they are glitched). I loaded a different level, and had the same problem, none of them made noise. i went back and reloaded the level, but this time unchecked the "online" box. Once I was playing offline, they started working again, it somehow fixed the glitch. Now they work whether I'm on or offline.

Is there a torch above it? If so destroy it and it should work.

Is there a block UNDER it? if not it won't work.

You should not put them on dirt. If you do will not work. The same thing happened with me.

This is a definite bug, as you seem to be doing everything right. It certainly isn't game breaking. What did you need it for, anyway?