Ultradiscretization is a limiting procedure by which a given difference equation is approximated by a piecewise linear equation, which can be considered as a time evolution rule of a cellular automaton. Although some attempts have been carried out to apply this technique to certain differential equations, no systematic procedure has been found transforming a given differential equation into an ultradiscrete equation. The paper under review constitutes a step forward along this direction, in the sense that ultradiscrete analogues are systematically built from first-order differential equations. The technique is applied to several reaction-diffusion partial differential equations, in particular, the Fisher and Allen-Cahn semilinear equations. Moreover, since the resulting ultradiscrete systems are piecewise linear, several solutions are constructed, in particular stationary solutions, traveling wave solutions and also entire solutions. The ultradiscretization procedure has in addition several favorable properties, such as the preservation of fixed points and the positivity of the solution.