But doesn't voltage play a part also?
The wire only "knows" about itself, and what it "knows" is exactly what you know, that it has a resistance, and that it's passing a current.
The voltage that the wire drops is defined therefore by Ohm's Law.
V=IR
You know I - the current you are passing (3A),
You know R the resistance of your wire (whatever it is, measure if you like),
Thus you calculate V, that being the voltage difference between one end of the wire and the other.
Voltage is a consideration in wire only in so far as insulation - high voltages can pierce insulation and establish a spark to lower potential things.