Routers do not change the MAC address, they completely strip off the layer-2 frame, losing any layer-2 information, e.g. MAC addressing. A router then routes the layer-3 packet that was encapsulated in the lost frame to a different interface, and it builds a new frame for the new interface.
Remember that not all layer-2 protocols use MAC addressing, only the IEEE protocols (ethernet, token ring, Wi-fi, etc.) do.. For example, frame relay uses DLCI, ATM uses VPI/VCI, and PPP does not use addressing. If the next interface is connected with one of those protocols, the frame the router builds will not have MAC addressing. It is common fot things like xDSL to use PPP and ATM, so there is no MAC addressing on the WAN interface.
On the other hand, switches are transparent devices that do not change the frame, so any MAC addressing is not disturbed.