Robust bubble-free priming of complex microfluidic chips represents a critical, yet often unmet prerequisite to enable their practical and widespread application. Towards this end, the usage of a network of capillary stop valves as a generic design feature is proposed. Design principles, numerical simulations, and their application in the development of a microfluidic cell culture device are presented. This chip comprises eight parallel chambers for the assembly and cultivation of human hepatocytes and endothelial cells. The inlet channel divides into cell chambers, after which the flows are reunited to a single chip outlet. Dimensions and geometry of channels and cell chambers are designed to yield capillary burst pressures sequentially increasing towards the chip outlet. Thus, progress of liquid flow through the device is predefined by design and enclosure of air bubbles inside the microfluidic structures is efficiently avoided. Capillary stop valves were designed using numerical simulations. Devices were fabricated in cyclic olefin polymer. Pressure during filling was determined experimentally and is in good agreement with data obtained from simulation.