The significance of drop-fluid viscosity on the effective rheological properties and on the dynamics of the microstructure of mono-disperse suspensions of two-dimensional liquid drops with constant interfacial tension is investigated by means of numerical simulations at vanishing Reynolds number, using the boundary integral method for Stokes flow. Three important features of the numerical method are the computation of the doubly-periodic Green's function and associated stress tensor by tabulation and interpolation, the iterative solution of a deflated integral equation for the interfacial velocity, and the repositioning of the drop interfaces at close proximity to avoid artificial coalescence. In the first part of the simulations, the interaction of two intercepting drops in simple shear flow is studied in an extended range of conditions, and the results are used to quantify the behaviour and develop insights into the physics of dilute systems. In the second part of the simulations, the motion of a random suspension of 25 drops repeated periodically in the two spatial directions is studied at the areal fraction ϕ=0.30, drop fluid to ambient fluid viscosity ratio λ=1 or 10, and drop capillary number Ca=0.10 or 0.30, a total of four combinations. It is found that the rheological properties of the suspension and the average drop deformation and orientation depend on the values of λ and Ca in a subtle fashion. As the viscosity of the drops is raised, the drop-centre pair distribution function undergoes a transition from a liquid-like to a rigid-particle-like behaviour, and particle aggregation and cluster formation become more important. For λ=10, the results are in excellent qualitative, and in some cases quantitative, agreement with those presented in previous studies for mono-layered suspensions of rigid spheres. The drop self-diffusivity is computed and its dependence on λ and Ca is discussed, although the results carry some uncertainty owing to the moderate number of drops within each periodic cell.