Long autoignition delay time and low lateral flame propagation speed are among the key problems in developing high-speed combustors for ram/scramjet engines. Plasma-assisted combustion can help to solve these problems. Estimates indicate that uniform volumetric nonequilibrium cold plasma ignition of fuel-air mixtures in ram/scramjet combustors can require large amounts of power to be deposited into the flow. Therefore, in this paper, we explore a possibility of increasing flame propagation speed that would be complementary to plasma ignition, allowing the latter to be applied to a smaller volume. The approach suggested in this paper is to use microwaves for flame speed enhancement. A rectangular microwave cavity was used to incorporate the burner which produces floated propane-air and methane-air flames. The microwave source is capable of providing microwave power in the range of 1300 W to 4500 W at 2.45 GHz. Successful experiments with laminar premixed flames at standard atmospheric pressure in a microwave resonant cavity have demonstrated the flame speed increase up to 68% in methane-air flame at the equivalence ration of 0.7 and up to 57% in propane-air flame at the equivalence ratio of 0.6. The microwave field in the experiments was subcritical, i.e., its intensity was below that required for breakdown, and a small amount of power (on the order 10 W) was absorbed in the flame. Theoretical estimates indicate that microwave absorption in the thin flame front, where radical reactions produce electrons and ions, can increase the flame temperature by up to a few hundred degrees Kelvin, resulting in a substantial flame speed increase with a relatively low absorbed power. These theoretical estimates support the observed experimental data.