Nuclear modulation in electron-spin-echo spectroscopy is conventionally studied by one-dimensional electron-spin-echo envelope modulation (ID-ESEEM). Two-dimensional Fourier transform electron-spin resonance (2D-FTESR) studies of nuclear modulation have the promise of enhancing the spectral resolution and clarifying the key details of the relaxation processes. We present a 2D-FTESR study on single proton nuclear modulation from γ-irradiated malonic acid single crystals to test the validity of the Gamliel-Freed theory and to assess the value of the new methods. The two pulse spin-echo correlation spectroscopy (SECSY) spectra as a function of orientation of the single crystal show very good agreement with the Gamliel-Freed theory extended to the general case of nonaxially symmetric hyperfine interaction. It is very simply affected by spin relaxation, such that relative intensities are essentially unaffected. Thus SECSY-ESR can most reliably be utilized for studying nuclear modulation. Stimulated SECSY provides the simplest nuclear modulation patterns, which, however, do exhibit the suppression effect well known in three-pulse ESEEM studies. Two-dimensional electron-electron double resonance (2D-ELDOR) provides nuclear modulation patterns similar to that of SECSY-ESR, so the suppression effect is absent. Both three-pulse methods exhibit complex relaxation behavior which can affect relative intensities. This is a feature characteristic of three-pulse ESEEM, but is not well understood. It is shown how the 2D-FTESR methods enable one to obtain the details of the complex spin relaxation, and in the process, obtain very good agreement between experiment and theory. 2D-ELDOR exhibits exchange cross peaks as well as coherence peaks from the nuclear modulation. It is shown how experiments, as a function of mixing time, enable one to separate the effects of the two. It is pointed out that such experiments are in the spirit of 3D spectroscopy. A new observation of the broadening of the 2D-ELDOR main peaks with an increase in mixing time is ascribed to the effects of solid-state dynamical processes that are slow on the ESR time scale and may thus be studied in "real time" in such experiments. The analysis of spin relaxation in this study is enabled by a full Liouville space derivation of the combined effects of nuclear modulation and spin relaxation in 2D-FTESR.