Propagating wave-like disturbances associated with solar flares - commonly observed in the chromosphere as Moreton waves - have been known for several decades. Recently, the phenomenon has come back into focus prompted by the observation of coronal waves with the SOHO/EIT instrument (``EIT waves"). It has been suggested that they represent the anticipated coronal counterpart to Moreton waves, but due to some pronounced differences, this interpretation is still being debated. We study 12 flare wave events in order to determine their physical nature, using Hα, EUV, He I 10 830 Å, SXR and radioheliographic data. The flare wave signatures in the various spectral bands are found to lie on closely associated kinematical curves, implying that they are signatures of the same physical disturbance. In all events, and at all wavelengths, the flare waves are decelerating, which explains the apparent ``velocity discrepancy" between Moreton and EIT waves which has been reported by various authors. In this paper, the focus of the study is on the morphology, the spatial characteristics and the kinematics of the waves. The characteristics of the common perturbation which causes the wave signatures, as well as the associated type II radio bursts, will be studied in companion Paper II, and a consistent physical interpretation of flare waves will be given.