The Spartans indeed did not have radio^[citation ^needed] and as far as we know the Greeks never used standards, bells, drums or other such fancy gear to convey commands in battle. Their command and control was almost bafflingly primitive.

A variety of signals were actually in use. Flags and fire signals were known in naval warfare; in exceptional circumstances, such signals could also be used on land. At the battle of Potidaia (432 BC), the defenders of the city arranged for allied cavalry to attack the Athenians in the rear at the raising of a banner on the city wall. In his treatise on the defence of cities, Aineias the Tactician recommends the use of fire signals to communicate orders to outposts and sallying forces. However, these signals relied on visibility from a distance; they could only be useful if they could be raised in high places (towers, hilltops) or on the open sea. In the confusion of pitched battle, they would be useless.

The Greeks therefore relied on an instrument that could be heard, and one that was loud enough to be heard even over the noise of battle: the salpinx, a long, tube-shaped trumpet. The phalanx was driven on by the deafening blasts of the salpinx. In Hellenistic times, the orders given with this instrument became quite sophisticated. The Classical Greeks, however, had neither the discipline nor the training to recognise and respond to complex signals in the heat of battle. If our sources are right, they actually used the salpinx for only two distinct signals: the charge and the retreat. More interestingly, only troops trained by Spartans appear to have known the second one. The rest of the Greeks knew the sound of the trumpet only as their cue to charge screaming into their enemies.

If more subtlety was required, and especially if the army was a large coalition force, messengers on horseback could be used to convey orders to allied commanders. At the battle of Plataia (479 BC), where the Athenian and Spartan wings were deployed several miles apart, horsemen brought messages back and forth between their respective generals. We don't get many examples of this, but it must have been common practice, given the sheer length of many Greek battle lines. On the other hand, subtlety and detailed commands were not the way of the Greeks. To keep things simple for their amateur armies, Greek generals very rarely gave fresh orders once the army was already drawn up for battle.

If there was an emergency that demanded a response, the only thing a general could really do was shout, and hope he would be heard. It was common practice for a watchword or creed to be passed down the line before battle, with every man shouting it to the next; orders could theoretically be given the same way. However, in the stress and chaos of battle, distracted by noise and imminent death, untrained militia could hardly be expected to beat anyone at a game of telephone. Orders given in battle tended to be given only to the general's immediate entourage. The rest of the army was effectively beyond his control.

The Spartans were the great exception to this rule. Unlike the forces of all other Greeks, their armies were not large mobs, but carefully arranged collections of units divided into sub-units and commanded by a hierarchy of officers. This system allowed them to do what no other Greek force could do: give orders to troops in battle. An admiring Thucydides (5.66.3-4) explains how it's done:

When a king is in the field, all commands proceed from him: he gives the word to the polemarchs; they to the lochagoi; these to the pentekostyes; these again to the enomotarchoi, and these last to the enomotiai. In short, all orders required pass in the same way and quickly reach the troops; as almost the whole Lakedaimonian army, save for a small part, consists of officers over officers, and the care of what is to be done falls upon many.

In short, they also shouted, but from superior to subordinate, until the lowest ranking officer shouted the order to his men. Other Greeks lacked this elaborated chain of command in which each officer knew what his responsibilities were. The same principle applied to both hoplites and horsemen:

When anything has to be done, the word of command is much more effective if it is passed to officers rather than to common men. Let us assume that this formation has been adopted: every file-leader must know his position in the line of march by word passed along by the wing commander, just as every wing commander is informed by the cavalry commander of his proper place in the charge. For when these instructions are given there will be much better order than if the men get in each other's way like a crowd leaving the theatre.

-- Xenophon, The Cavalry Commander 2.6-7

In this way, even though they used no technological means whatsoever, the Spartans could still be structurally better at battlefield command and control than their rivals. Apparently, it sufficed; even the later Hellenistic pike phalanxes appear to have relied mostly on vocal commands to manoeuvre their carefully subdivided bodies of men.