Engineering ground-motion parameters can be used to describe the damage potential of an earthquake. Some of them correlate well with several commonly used demand measures of structural performance, liquefaction, and seismic-slope stability. The importance of these parameters comes from the necessity of an alternative measure to the earthquake intensity. In the proposed new attenuation relationship we consider peak values of strong motion, spectral acceleration, elastic input energy at selected frequencies, root-mean-square acceleration, Arias intensity, characteristic intensity, Fajfar index, cumulative absolute velocity, cumulative absolute velocity integrated with a 5 cm/sec2 lower threshold, and spectrum-intensity energy. This article describes the steps involved in the development of new attenuation relationships for all the preceding parameters, using all existing, up-to-date Greek strong-motion data. The functional form of the empirical equation is selected based on a theoretical model, and the coefficients of the independent variables are determined by employing mixed effects regression analysis methodologies.