In this paper, we present an extensive experimental study on the dynamic mechanical properties of composites with polymer matrices, as well as a quantification of the parameters that influence these properties. Polymer-composite matrices make it possible to form any reinforcement arrangement of fibres, particles, and layers, which makes it possible to form composite materials with certain dominant mechanical properties according to the internal arrangement for the application. In this study, we focused on the dynamic properties (i.e., damping parameters, such as the loss factor (tan d), logarithmic decrement (λ), storage modulus (E′), and loss modulus (E″)) of composites with polymer matrices, including parameters such as the fibre material, fabric weaving, fibre orientation, temperature, frequency, particle size, volume of short fibres, and epoxy resin type. If other articles focus on one type of composite and 1–2 parameters, then the benefit of this article lies in our analysis of 8 mentioned parameters in the experimental analysis of 27 different types of composites with polymer matrices. The tested fibre materials were glass, aramid, and carbon; the tested woven fabrics were twill, plain, unidirectional, and satin; the temperature range was from −50 to +230 °C; the frequency was 1 Hz and 10 Hz; the particle size was 0.1–16 mm; the volume percentages of the short fibres were 3, 6, and 12 vol.% of the hybrid polymer composites and the type of polymer matrix. We used the free-damped-vibration method with vibration dynamic signal analysis and the forced-damped vibration of dynamic mechanical thermal analysis for testing. We ranked the parameters that influence the dynamic vibration properties according to the effects. Among sets of results provided in the paper, considering the storage modulus, loss modulus, and loss factor, the best results of the fibre composites were for aramid-fibre-reinforced polymers, regardless of the weave type, with an advantage for unidirectional fabric. The best results of the particle composites were for those with fine filler sizes that incorporated the short fibres.