The scattering of orthopositronium from He, Ne, Ar, Li+, Na+, and K+ is investigated with the fixed-core stochastic variational method. The scattering length for Ps-He scattering was 1.57a(0), a value consistent with the threshold cross sections derived from three positron lifetime experiments. The scattering lengths for the Ps-Ne and Ps-Ar systems were 1.55a(0) and 1.79a(0), respectively. That there was only a 15% variation among the scattering lengths is compatible with the experiment of Coleman et al. [J. Phys. B 27, 981 (1994)], who found the low-energy cross sections for He, Ne, and Ar to be about the same size. The scattering lengths for Ps scattering from Li+, Na+, and K+ were 12.9a(0), 28.5a(0), and -1.9a(0). The relatively small magnitude of the Ps-K+ scattering length strongly supports previous suggestions that the Ps-K+ system does not support a bound state. The annihilation parameter (1)Z(eff) has also been computed as part of the analysis. The present values of (1)Z(eff) are about 2.5-3 times smaller than the accepted experimental values since short-range electron-positron correlations were not taken into consideration when the annihilation matrix element was evaluated.