The answer ought to be a qualified "Yes." We don't only hear what we want, in as much as motivation has zero direct control over the transduction of auditory signals in a sensory sense. Motivation affects which way we turn our ears and whether we keep them in a room with sounds we want to hear vs. a room with sounds we don't want to hear, but if those sounds hit our ears, whether we want to hear them shouldn't affect whether the stereocilia in our cochleae wiggle and tell our brains all about it. That's the only qualification of the otherwise plainly affirmative answer that occurs to me.
In a perceptory sense, motivation matters a lot more, because transfer from echoic memory to working memory is a matter of attention, and motivation certainly affects attention. E.g., if you're listening for a particular sound, you'll notice it better (hence the cocktail party effect), and you'll even experience more false alarms in the signal detection sense. A personal example within the general example: all the sounds that had the same pitch and tone as the first note of my cell phone ringtone used to drive me nuts when I would take a shower while anticipating a call from a girl I really wanted to talk to, who would often call me unexpectedly and complain when I didn't answer.
Transfer from working memory to long-term memory is arguably even more sensitive to motivation, because it depends greatly on elaborative rehearsal. Hence the more obsessed (or just interested) you are with a particular topic is the more often and deeply you'll think about it, and the more likely you'll remember it after your working memory's fairly short duration is up. If you don't remember something, you can't say whether you heard it, so the better you remember is also the more evidence you can offer that you have heard it. This is important in an empirical epistemic sense for our purposes, partly because it exacerbates the effect as perceived retrospectively, and thus reinforces the folk theory in your title.