The rheology of confined flowing suspensions, such as blood, depends upon the dynamics of the components, which can be particularly rich when they are elastic capsules. Using boundary integral methods, we simulate a two-dimensional model channel through which flows a dense suspension of fluid-filled capsules. A parameter of principal interest is the equilibrium membrane perimeter, parameterized by xi (o), which ranges from round capsules with xi (o)=1.0 to xi (o)=3.0 capsules with a dog-bone-like equilibrium shape. It is shown that the minimum effective viscosity occurs for xi (o)a parts per thousand 1.6, which forms a biconcave equilibrium shape, similar to a red blood cell. The rheological behavior changes significantly over this range; transitions are linked to specific changes in the capsule dynamics. Most noteworthy is an abrupt change in behavior for xi (o)a parts per thousand 2.0, which correlates with the onset of capsule buckling. The buckled capsules have a more varied orientation and make significant rotational (rotlet) contributions to the capsule-capsule interactions.