We just bought a house with aluminum wiring, nothing is up to code (old house), so we are rewiring the whole house. We plan to construct a laundry room in the center of the house and we would like to locate the new service panel there, as there is no other obvious place to put it.

Apart from the 3 foot clearance for panel access, do we need a certain amount of clearance to the washing machine?

You need a min of 30 inch wide at the panel, and the three feet out in front of the panel. Oterwise you should have no problems. The washer can be right along side the edge of the 30 in wide space.