In this paper, a new three-dimensional modelling approach is described for studying fluid-viscoelastic cell interaction, the subcellular element Langevin (SCEL) method, with cells modelled by subcellular elements (SCEs) and SCE cells coupled with fluid flow and substrate models by using the Langevin equation. It is demonstrated that: (i) the new method is computationally efficient, scaling as (N) for N SCEs; (ii) cell geometry, stiffness and adhesivity can be modelled by directly relating parameters to experimentally measured values; (iii) modelling the fluid-platelet interface as a surface leads to a very good correlation with experimentally observed platelet flow interactions. Using this method, the three-dimensional motion of a viscoelastic platelet in a shear blood flow was simulated and compared with experiments on tracking platelets in a blood chamber. It is shown that the complex platelet-flipping dynamics under linear shear flows can be accurately recovered with the SCEL model when compared with the experiments. All experimental details and electronic supplementary material are archived at http://biomath.math.nd.edu/scelsupplementaryinformation/.