Strong stability preserving (SSP) Runge-Kutta methods are often desirable when evolving in time problems with components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit Runge-Kutta methods have very restrictive time steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time stepping methods for problems with a stiff linear component and a non-stiff nonlinear component. This work defines strong stability properties of integrating factor Runge-Kutta methods. It shows that it is possible to define integrating factor Runge-Kutta methods that preserve the desired strong stability properties satisfied by each of the components when coupled with forward Euler time stepping, or even given weaker conditions. Sufficient conditions are defined for integrating factor one-step and two-step Runge-Kutta methods to be SSP, namely that they are based on explicit SSP Runge-Kutta methods with non-decreasing abscissas or with operators replaced by the downwinded operator when the abscissas decrease. We find such one-step methods of up to fourth order and two-step methods up to eighth order with non-decreasing abscissas, analyze their SSP coefficients, prove their optimality in a few cases, and investigate downwinding approaches to preserve strong stability. These methods are tested to demonstrate their convergence, that the SSP time step predicted by the theory is generally sharp, and that the non-decreasing abscissa condition or downwind modification is needed to guarantee strong stability. Finally, this research shows that on typical total variation diminishing linear and nonlinear test problems our new SSP integrating factor Runge-Kutta methods outperform the corresponding explicit SSP Runge-Kutta methods, implicit-explicit SSP Runge-Kutta methods, and some well-known exponential time differencing methods.