The calculation of binding free energies is important in many condensed matter problems. Although formally exact computational methods have the potential to complement, add to, and even compete with experimental approaches, they are difficult to use and extremely time consuming. We describe a Grid-based approach for the calculation of relative binding free energies, which we call Steered Thermodynamic Integration calculations using Molecular Dynamics (STIMD), and its application to Src homology 2 (SH2) protein cell signalling domains. We show that the time taken to compute free energy differences using thermodynamic integration can be significantly reduced: potentially from weeks or months to days of wall-clock time. To be able to perform such accelerated calculations requires the ability to both run concurrently and control in realtime several parallel simulations on a computational Grid. We describe how the RealityGrid computational steering system, in conjunction with a scalable classical MD code, can be used to dramatically reduce the time to achieve a result. This is necessary to improve the adoption of this technique and further allows more detailed investigations into the accuracy and precision of thermodynamic integration. Initial results for the Src SH2 system are presented and compared to a reported experimental value. Finally, we discuss the significance of our approach.