There are several rules and reasons that should be considered. An earth ground is primarily needed for safety purposes. Safety means that any high voltage fault current is directly conducted to earth ground through the lowest possible resistance. The scenario that we are protecting against is when a fault inside your amp connects one side of the AC line to your chassis. We don't want a person to be able to touch a chassis that has high voltage on it, so we short that chassis to earth ground, effectively to the lowest voltage point in the AC line system. This connection is made through your line cord and the third prong on the plug (in North America). By electrical code, this third prong is connected via an independent bond wire to a ground rod next to your building.
An earth ground may be required for other reasons. It is necessary in some systems to establish a common reference for multiple pieces of equipment. For correct operation of multi-box systems whose interconnections are not balanced, the boxes must share a common circuit reference. Sometimes this connection is made through the earth ground of the AC line input.
Earth ground is sometimes also needed for RF grounding, and for high voltage transient protection, such as when lightning strikes an element of the system.
Other than these reasons, the typical circuit will work without having any connection to earth ground. Most simple circuits show a ground symbol on their schematics, a symbol which refers only to a circuit common reference. For most of your amp connections with the three lines in triangular form, the connection is made to circuit common (ie. 0 volts), not to earth ground. Typically in audio systems there is no functional need to connect circuit common to earth ground.
However, if you have chosen to connect your chassis to your circuit common (0 volts), then of course your 0 volts will also be connected directly to earth ground. Many items of consumer equipment have a chassis that is earth grounded but with no connection between circuit common (0 volts) and that chassis. In many cases, safety is realized by using a non-conductive chassis, or by double-insulating between the exposed chassis and any potential fault points internally. I think most power tools nowadays use the double-insulation principle and then have only two prongs on their plug.
Any further detailed answer requires that you post your schematic.