We demonstrate that momentum dependent nuclear interactions (MDI) have a large effect on the dynamics and on the observables of high energy heavy ion collisions : Pion and kaon yields are strongly suppressed by MDI. This effect is much stronger than the supression due to a hard local potential. which introduces an ambiguity into the recent extraction of the nuclear equation of state from pion (and kaon) yields. Also the transverse rnomentum distributions exhibit a strong dependence on the MDI. The collective flow angles and the deuteron to proton ratios, on the other hand, are rather insensitive to the MDI. They may therefore be related directly to the nuclear equation of state at high density. Similtaneous measurements of these observables in heavy ion collisions can give clues on the nuclear equation of state at densities of interest for supernova collapse and neutron star stability.