Should sentences like "I wonder..." end with a question mark?

Your hunch is correct. Bryan Garner writes:

Writers sometimes err by putting a question mark after an indirect question, especially one beginning with I wonder.

^{Garner's Modern American Usage}

If you are asking a question, then yes. If you are simply telling people what you're wondering about, then it isn't a question and it should not have a question mark.

Some examples to help clarify:

I often wonder about the future.

I wonder what is for breakfast.

I wonder... who is that?

People will often say phrases like these as asking a question but its usage can be considered awkward for written English.

I am curious about your name?

That is pie?

And you are?

The question marks here are more flagging the intonation we use in everyday speech. You can also use an ellipsis:

And you are...

Your name is...

On tests you will see a blank line:

The answer is _____________

But for "I wonder..." I would stick to a period/full stop. When you say it out loud, feel free to say it much like you ask a question.