I ended up following @vicenteherrera's approach, with some tweaks (that are possibly bootstrap 3 specific). 
Basically; we can't break trs, or tds because they're not block-level elements. So we embed divs into each, and apply our page-break-* rules against the div. Secondly; we add some padding to the top of each of these divs, to compensate for any styling artifacts.
<style>
    @media print {
        /* avoid cutting tr's in half */
        th div, td div {
            margin-top:-8px;
            padding-top:8px;
            page-break-inside:avoid;
        }
    }
</style>
<script>
    $(document).ready(function(){
        // Wrap each tr and td's content within a div
        // (todo: add logic so we only do this when printing)
        $("table tbody th, table tbody td").wrapInner("<div></div>");
    })
</script>
The margin and padding adjustments were necessary to offset some kind of jitter that was being introduced (by my guess - from bootstrap). I'm not sure that I'm presenting any new solution from the other answers to this question, but I figure maybe this will help someone.