We consider the optimal design of controlled experimental epidemics or transmission experiments, whose purpose is to inform the practitioner about disease transmission and recovery rates. Our methodology employs Gaussian diffusion approximations, applicable to epidemics that can be modeled as density-dependent Markov processes and involving relatively large numbers of organisms. We focus on finding (i) the optimal times at which to collect data about the state of the system for a small number of discrete observations, (ii) the optimal numbers of susceptible and infective individuals to begin an experiment with, and (iii) the optimal number of replicate epidemics to use. We adopt the popular D-optimality criterion as providing an appropriate objective function for designing our experiments, since this leads to estimates with maximum precision, subject to valid assumptions about parameter values. We demonstrate the broad applicability of our methodology using a diverse array of compartmental epidemic models: a time-homogeneous SIS epidemic, a time-inhomogeneous SI epidemic with exponentially decreasing transmission rates and a partially observed SIR epidemic where the infectious period for an individual has a gamma distribution.