Terrain classification is an important feature for walking robots because it allows the robots to stably move and operate on the terrain. Different terrain classification techniques have been developed. The techniques include the use of different exteroceptive and proprioceptive sensors with different classification methods. Whereas these techniques have been widely used to classify flat, hard, and rough terrains, their application to soft terrains has not been fully addressed. Achieving soft-terrain classification will expand the operational range of walking robots. Thus, in this study, we propose a new technique to classify various terrains including soft ones. The technique exploits haptic feedback (expressed only through ground contact force measurement of a legged robot) and neurodynamics with the temporal memory of a reservoir computing-based recurrent neural network. We used six different terrains to evaluate the performance of the proposed technique. The terrains include sand (loose ground), foams with different softness levels (soft ground), and floor (hard ground). The experimental results show that we can successfully classify all terrains with an accuracy of above 70%. Furthermore, owing to the temporal memory of the network, if the haptic feedback is transiently missing, the network will be still be able to classify the terrain considerably well.