How to change analog sensor signal to appropriate levels for microcontrollers?
Yes, you can do this with a resistor divider.  We usually think of a resistor divider as multiplying a voltage by some value less than 1, meaning scaling relative to ground.  However, a resistor divider can be set up to scale relative to any particular voltage.

In your case, you can scale the voltage about the 3.3 V supply instead of about ground.  Overall, you want to scale a 6 V range down to a 3 V range, so the divider scale factor needs to be 1/2, meaning the two resistors must be equal.  For example, one 10 k&Omega; resistor to the 3.3 V supply, another to the &plusmn;3 V signal, and the two other ends connected to the A/D input.  That will load the signal by the sum of the resistors (20 k&Omega;) and provide a output impedance for driving the A/D by the parallel combinations of the two resistors (5 k&Omega;).  If 5 k&Omega; is not low enough to drive your A/D, then lower the two resistor values so that each is twice the required impedance.

Note that this method requires that the 3.3 V supply is well regulated.  That would be the case, for example, if it is driven by a linear regulator from a higher voltage.