What is the earliest evidence of soldiers, or armies, marching in step?

There are reliefs and depictions of heavy infantry advancing in formation from ancient Egypt and Mesopotamia, dating back to the second millennium BC. They show rows of shields and spears, and men moving in unison. These are probably units marching in step - but we don't have written evidence to confirm it, and the images might just show stylised attacks by bodies of men.

There are 7th century BC vase paintings of Greeks advancing into battle in neat lines, but modern scholars doubt that these are evidence of formation drill. More likely they are just an attempt by the painter to show lots of men charging together. Some of the men run while others walk; all are armed with javelins as well as thrusting spears. These lines only occur on a couple of known pots, all by the same painter, and never reappear in later times.

The earliest unambiguous evidence is Thucydides' description of the Spartans (here referred to as Lakedaimonians) at the First Battle of Mantineia, 418 BC:

After this they joined battle, the Argives and their allies advancing with haste and fury, the Lakedaimonians slowly and to the music of many flute-players—a standing institution in their army, that has nothing to do with religion, but is meant to make them advance evenly, stepping in time, without breaking their order, as large armies often do when they go into battle.

-- Thuc. 5.70

The interesting thing about this description is that Thucydides felt a need to explain what the Spartans were doing. Clearly he expected his readers to have no idea what the flute-players were for. His account shows that the Spartans were the only Greeks who knew how to march in step. In his recommendations on training an army, the veteran mercenary general Xenophon does not expect a commander to train his men to do this. It got in the way of the troops' desire to get at the enemy, and even the Spartans sometimes struggled to maintain their discipline:

It is reported that the soldiers were inspired with so much strength and courage that it was a task for their leaders to restrain them as they pushed forward to the front.

-- Xenophon, Hellenika 7.1.31

It wasn't until the rise of the Macedonian pike phalanx that marching in step really became an essential feature of infantry drill. Where hoplites could just about make do without it, a phalangite was useless unless he was part of a well-trained unit.