Solitons forming inside optical fibers are perturbed by several higher-order dispersive and nonlinear effects, especially when ultrashort optical pulses are used to excite them. We study, both analytically and numerically, how the radiation emitted by solitons in the form of dispersive waves (sometimes called Cherenkov radiation) is affected by these higher-order effects. Our results show that a certain minimum amount of third-order dispersion is needed before the amplitude of the dispersive wave becomes large enough for a spectral peak to appear in the output spectrum. This minimum value depends on the soliton order N and decreases with increasing N. The amplitude of the radiation peak increases initially with both the magnitude of the third-order dispersion and the soliton order, but then saturates to a relative power level that is typically below 10% of the launched power. Our results reveal several interesting features that should be of relevance for applications requiring an ultrabroadband optical source.