Pulsar radio emission involves a coherent emission mechanism involving relativistic particles, but there is no consensus on what the specific emission mechanism is. There are arguments for and against several suggested mechanisms of long-standing, including two, relativistic plasma emission and anomalous Doppler emission, that depend explicitly on the dispersive properties of waves in a pulsar plasma. Models that do not include dispersion due to the relativistic spread in energy of the plasma particles can be misleading, and including the effects of this spread on the wave dispersion leads to severe constraints on these two suggested mechanisms. It is argued that all suggested mechanisms are implausible. A variant of relativistic plasma emission involving large-amplitude oscillations needs to be explored as a possible alternative.