We're not to the point of an echocardiogram yet, but we have ruled out taurine deficiency (he's been on a high quality prescription diet chock full of taurine for a year now), hypertension, and a couple of infections that could cause it.

I'm suspecting it's weight/thyroid related. We've got him on a diet now, but while hypothyroidism is not so common in cats, he's gained over a pound in the six months since his last vet visit, so it's possible.