The simplest is an infusion-based method. Since you start from whole beans, you will have to grind them, but when that's done it's just steeping them in water. According to Wikipedia on coffee maker history:
For hundreds of years, making a cup of coffee was a simple process. Roasted and ground coffee beans were placed in a pot or pan, to which hot water was added, followed by attachment of a lid to commence the infusion process. Pots were designed specifically for brewing coffee, all with the purpose of trying to trap the coffee grounds before the coffee is poured. Typical designs feature a pot with a flat expanded bottom to catch sinking grounds and a sharp pour spout that traps the floating grinds. Other designs feature a wide bulge in the middle of the pot to catch grounds when coffee is poured.
The above is really simple, it's just four simple steps:
Grind the beans. The quality of the grinder and choosing the correct grind setting will influence the result a lot.
Add ground beans and hot water to a pot. Water should be just of the boil, pour water in the container first to heat the pot, similar to the process for a French press.
Wait until the infusion has reached the desired level. Again, basically the same as the French press method.
Pour the coffee from the pot into a cup leaving most of the grounds in the pot.
Alternatively, the fourth step can be replaced by straining the contents of the pot in a metal or paper filter.
This method has most of the benefits of the French press with less cleaning up because there is no built in strainer. You will have to clean your cup, the pot and the metal strainer or paper filter holder (if using), but depending on what specific equipment you use these can all be cleaned in the dishwasher.