My favourite place to shop in Budapest is Vámház körúti Vásárcsarnok, although i love all the markets (piac) in Hungary. The prices are great, probably not as cheap as the major chains, but the produce is a hell of lot fresher than Tesco.
U.S. does not have these kinds of markets in every neighbourhood, which is surprising to me. In Lyon, there are markets in every neighbourhood, but they are portable and not every day.