Continued scaling-down of lithographic-pattern feature sizes has brought templated self-assembly of block copolymers (BCPs) into the forefront of nanofabrication research. Technologies now exist that facilitate significant control over otherwise unorganized assembly of BCP microdomains to form both long-range and locally complex monolayer patterns. In contrast, the extension of this control into multilayers or 3D structures of BCP microdomains remains limited, despite the possible technological applications in next-generation devices. Here, we develop and analyse an orthogonal self-assembly method in which multiple layers of distinct-molecular-weight BCPs naturally produce nanomesh structures of cylindrical microdomains without requiring layer-by-layer alignment or high-resolution lithographic templating. The mechanisms for orthogonal self-assembly are investigated with both experiment and simulation, and we determine that the control over height and chemical preference of templates are critical process parameters. The method is employed to produce nanomeshes with the shapes of circles and Y-intersections, and is extended to produce three layers of orthogonally oriented cylinders.