Within the framework of continuum fluid dynamics, couple-stresses appear as an inevitable consequence of non-central forces and the discrete character of matter at the finest scales. As a result, the force-stress tensor becomes non-symmetric and classical theory may not accurately predict the behavior. Recent theoretical work has shown that the couple-stress tensor must be skew-symmetric and that mean curvature rate is the energy-conjugate kinematical measure. The resulting fully consistent couple stress theory incorporates a characteristic material length scale for the fluid that becomes increasingly important, as the characteristic geometric dimension of the problem approaches that level. This size-dependent non-Newtonian theory is essential to understand a range of behavior at micro-scales with potential applications to blood flow and lubrication, among others. Here we concentrate on steady creeping flow within this newly developed fully determinate linear couple stress theory and formulate a boundary integral representation for two-dimensional problems. These boundary integral equations are written in terms of velocities, angular velocities, force-tractions and moment-tractions as primary variables. Details are provided on the derivation of fundamental solutions and on the corresponding boundary element implementation. Afterwards, the new boundary element method is applied for the solution of two basic problems to explore some consequences of this size-dependent couple stress fluid mechanics. (C) 2013 Elsevier B.V. All rights reserved.