Laser-induced acoustic desorption (LIAD) from thin metal foils is a promising technique for gentle and efficient volatilization of intact organic molecules from surfaces of solid substrates. Using the single-photon ionization method combined with time-of-flight mass spectrometry, we have examined the neutral component of the desorbed flux in LIAD and compared it to that from direct laser desorption. These basic studies of LIAD, conducted for molecules of various organic dyes (rhodamine B, fluorescein, anthracene, coumarin, BBQ), have demonstrated detection of intact parent molecules of the analyte even at its surface concentrations corresponding to a submonolayer coating. In some cases (rhodamine B, fluorescein, BBQ), the parent molecular ion peak was accompanied by a few fragmentation peaks of comparable intensity, whereas for others, only peaks corresponding to intact parent molecules were detected. At all measured desorbing laser intensities (from 100 to 500 MW/cm2), the total amount of desorbed parent molecules depended exponentially on the laser intensity. Translational velocities of the desorbed intact molecules, determined for the first time in this work, were of the order of hundreds of meters per second, less than what has been observed in our experiments for direct laser desorption, but substantially greater than the possible perpendicular velocity of the substrate foil surface due to laser-generated acoustic waves. Moreover, these velocities did not depend on the desorbing laser intensity, which implies the presence of a more sophisticated mechanism of energy transfer than direct mechanical or thermal coupling between the laser pulse and the adsorbed molecules. Also, the total flux of desorbed intact molecules as a function of the total number of desorbing laser pulses, striking the same point on the target, decayed following a power law rather than an exponential function, as would have been predicted by the shake-off model. To summarize, the results of our experiments indicate that the LIAD phenomenon cannot be described in terms of simple mechanical shake-off or direct laser desorption. Rather, they suggest that multistep energy-transfer processes are involved. Two possible (and not mutually exclusive) qualitative mechanisms of LIAD that are based on formation of nonequilibrium energy states in the adsorbate-substrate system are proposed and discussed.