We consider an extention of the familiar Schur product to a bilinear product on the space of matrices whose entries are either bounded operators on a fixed Hilbert space or bounded "square" operator matrices. We show that this is a "natural" non-commutative extention of the Schur product, which retains many of its properties. The work is done mainly in infinite dimensions, where we concentrate on the maps induced on the space of bounded operator matrices via left or right "Schur block-multiplication" by a fixed "Schur block-multiplier". Our main goal is to study the distinctions between left and right multipliers, as well as the behaviour of ideals of operators under action of maps induced bu such.