OLD white-rodgers thermostat
All you need is two wires:R and W. These will give you heat. You could run 4 wires total, if you so desired. In addition to R and W, you could hook up C, to power the new stat, and G, to let you control the fan (off/on/auto). What your thermostat does now is turn your "very fancy furnace" on when heat is needed, and off when it reaches the desired temperature. It basically acts as a switch between the two wires. When heat is needed, it closes the connection, and when heat is no longer required, it opens the connection, thereby turning off the heat.