In simulating continuum model fluids that undergo phase separation and criticality, significant gains in computational efficiency may be had by confining the particles to the sites of a lattice of sufficiently fine spacing, a(0) (relative to the particle size, say a). But a cardinal question, investigated here, then arises; namely, How does the choice of the lattice discretization parameter, zeta equivalent to a/a(0), affect the values of interesting parameters, specifically, critical temperature and density, T-c and rho(c)? Indeed, for small zeta (<= 4-8) the underlying lattice can strongly influence the thermodynamic properties. A heuristic argument, essentially exact in d = 1 and d = 2 dimensions, indicates that, for models with hard-core potentials, both T-c(zeta) and rho(c)(zeta) should converge to their continuum limits as 1/zeta((d+1)/2) for d <= 3 when zeta -> infinity but the behavior of the error is highly erratic for d >= 2. For smoother interaction potentials, the convergence is faster. Exact results for d = 1 models of van der Waals character confirm this; however, an optimal choice of can improve the rate of convergence by a factor 1/zeta. For d >= 2 models, the convergence of the second virial coefficients to their continuum limits likewise exhibits erratic behavior, which is seen to transfer similarly to T-c and rho(c); but this can be used in various ways to enhance convergence and improve extrapolation to as is illustrated using data for the restricted primitive model electrolyte.