X-ray echo spectroscopy, a space-domain counterpart of neutron spin-echo, is a recently proposed inelastic x-ray scattering (IXS) technique. X-ray echo spectroscopy relies on imaging IXS spectra, and does not require x-ray monochromatization. Due to this, the echo-type IXS spectrometers are broad-band, and thus have a potential to simultaneously provide dramatically increased signal strength, reduced measurement times, and higher resolution compared to the traditional narrowband scanning-type IXS spectrometers. The theory of x-ray echo spectrometers presented in [1] is developed here further with a focus on questions of practical importance, which could facilitate optical design and assessment of the feasibility and performance of the echo spectrometers. Among others, the following questions are addressed: spectral resolution, refocusing condition, echo spectrometers tolerances, refocusing condition adjustment, effective beam size on the sample, spectral window of imaging and scanning range, the impact of the secondary source size on the spectral resolution, the angular dispersive optics, the focusing and collimating optics, etc. Examples of optical designs and characteristics of echo spectrometers with 1-meV and 0.1-meV resolutions are presented.