One thing the parent didn't bring up was there was a philosophical movement called "rationalism" that preceded the contemporary movement that refers itself by the same name. These so-called "Continental Rationalists" included philosophers like Descartes and Leibniz. The British Empiricist movement, featuring Locke, Berkeley and Hume, arose in response to the Continental Rationalists.

One of the key points of contention between these two movements was the problem of "innate knowledge". Rationalists believed that we were born with some sorts of knowledge (foundationalism). Leibniz believed that we were born with the basic mathematical knowledge and reasoning. Descartes thought he had a great deal of innate ideas -- perhaps most famously, his proof of the existence of God relied on the fact that he had an innate idea of God.

It was really only until Locke came along that anyone presented a top-down theory attempting to account for the acquisition of basic forms of knowledge (e.g. difference and similarity, truth and falsehood) while maintaining an initial "tabula rasa" state.

I wouldn't lump in the contemporary rationalist movement with either of these previous movements -- it is probably best considered as an extension of logical positivism, owing to heirs like early Wittgenstein.

However, positivism brushed up against many philosophical problems: semantics, the analytic-synthetic dichotomy, etc. Things got worse when Wittgenstein eventually wrote "Philosophical Investigations" which effectively contradicted everything he wrote in the "Tractatus Logico-Philosophicus", the latter of which was the foundational text of logical positivism. The two schools of thought were so different that people generally refer to "Tractatus" and "post-Tractatus" Wittgenstein.

One of the key scientific critics to logical positivism was Karl Popper -- best known for formalizing the ideas of falsifiability -- eschewed the idea of induction altogether. I highly recommend his "The Logic of Scientific Discovery" (which is not to say that I agree with Popper, but he is an important figure in the philosophy of science).

Thomas Kuhn's "The Structure of Scientific Revolutions" calls into question the very idea of there being a "scientific truth" that is independent of historical assumptions. Contemporary heirs of this kind of historically focused thought, like Ian Hacking, try to make room for the ideas of "scientific fact" in some contexts (particle physics) but question it in others (psychology). I highly recommend Hacking's "Making Up People" -- a brilliant collection of essays which is still friendly to people who haven't read a great deal of Foucault (another important figure in recent philosophy of science).

Second to last in my brief and certainly not comprehensive list is Hilary Putnam. It's hard to precisely pin down Putnam's thought -- as a philosopher he is remarkably open to criticism -- he has written papers in response to himself. Putnam has mostly kept himself consistent with scientific realism -- which basically states that scientific theories are usually more-or-less true.. Early on, he was a metaphysical realist who became sharply opposed to the school. Frankly, I don't know where he stands today, but read his essay "Brains in a vat" if you're at all interested in Skepticism.

Finally, this list of alternatives to positivism would be incomplete without Richard Rorty. His landmark text, "Philosophy and the Mirror of Nature", rejects the possibility of an epistemology altogether. I can't recommend Rorty's writings enough -- I believe he stands with Foucault as the most important pair of philosophers in the last 50 years.

I hope that I've given some more background on the rationalist/empiricist debate and contemporary alternatives to proper empiricist and positivist thought. I certainly cannot give you anything comprehensive in the space of a HN comment -- my only goal has been to hint towards various schools of thought on this matter. Pierre Hadot wrote that philosophy is a way of life moreso than a domain of thought -- and I tend to believe that in life, the journey is more important than the destination. In that respect, it is impossible for me to give you a firm answer to the question you asked of the OP: "if not empiricism, then what?" -- but I hope that my comments help you find where you stand.