We use confocal microscopy to directly visualize the formation and complex morphologies of trapped non-wetting fluid ganglia within a model 3D porous medium. The wetting fluid continues to flow around the ganglia after they form; this flow is characterized by a capillary number, Ca. We find that the ganglia configurations do not vary for small Ca; by contrast, as Ca is increased above a threshold value, the largest ganglia start to become mobilized and are ultimately removed from the medium. By combining our 3D visualization with measurements of the bulk transport, we show that this behavior can be quantitatively understood by balancing the viscous forces exerted on the ganglia with the pore-scale capillary forces that keep them trapped within the medium. Our work thus helps elucidate the fluid dynamics underlying the mobilization of a trapped non-wetting fluid from a 3D porous medium.