The linear ballistic accumulator model is a theory of decision-making that has been used to analyse data from human and animal experiments. It represents decisions as a race between independent evidence accumulators, and has proven successful in a form assuming a normal distribution for accumulation (“drift”) rates. However, this assumption has some limitations, including the corollary that some decision times are negative or undefined. We show that various drift rate distributions with strictly positive support can be substituted for the normal distribution without loss of analytic tractability, provided the candidate distribution has a closed-form expression for its mean when truncated to a closed interval. We illustrate the approach by developing three new linear ballistic accumulation variants, in which the normal distribution for drift rates is replaced by either the lognormal, Fréchet, or gamma distribution. We compare some properties of these new variants to the original normal-rate model.