The linear thermal expansion for a number of BCC solid solution alloys in the system Zr-Nb-Mo-Re has been measured at room temperature and at temperatures between 1.5 and 35K. Values of the room temperature Gruneisen parameter gamma G(300) and the low temperature lattice, gamma l, and electronic, gamma e, Gruneisen parameters have been derived from the expansion data. These parameters have been applied to a discussion of model calculations for the pressure dependence of the superconducting transition temperature in this alloy system.