Ecobee 3 - I have 3 Wires
Unfortunately, your old thermostat likely did not have power going to it,  just sense lines that closed a circuit when the mercury switch attached to the thermostat closed.  All of the &quot;cool&quot; new thermostats require power and a neutral line (so minimum 4 wires).
Is there any way you can get power to the planned install location? they make 24V plug in transformers to do this, here is one transformer option;  then you can use the existing wires as needed assuming they are get-at-able from both sides.