Three.js png texture – alpha renders as white instead as transparent

the opacity attribute of material does the trick for you. Follows, example code snippet:

var materialArray = [];

materialArray.push(new THREE.MeshBasicMaterial( { map: THREE.ImageUtils.loadTexture( 'images/xpos.png' ), transparent: true, opacity: 0.5, color: 0xFF0000 }));
materialArray.push(new THREE.MeshBasicMaterial( { map: THREE.ImageUtils.loadTexture( 'images/xneg.png' ), transparent: true, opacity: 0.5, color: 0xFF0000 }));
materialArray.push(new THREE.MeshBasicMaterial( { map: THREE.ImageUtils.loadTexture( 'images/ypos.png' ), transparent: true, opacity: 0.5, color: 0xFF0000 }));
materialArray.push(new THREE.MeshBasicMaterial( { map: THREE.ImageUtils.loadTexture( 'images/yneg.png' ), transparent: true, opacity: 0.5, color: 0xFF0000 }));
materialArray.push(new THREE.MeshBasicMaterial( { map: THREE.ImageUtils.loadTexture( 'images/zpos.png' ), transparent: true, opacity: 0.5, color: 0xFF0000 }));
materialArray.push(new THREE.MeshBasicMaterial( { map: THREE.ImageUtils.loadTexture( 'images/zneg.png' ), transparent: true, opacity: 0.5, color: 0xFF0000 }));

var MovingCubeMat = new THREE.MeshFaceMaterial(materialArray);
var MovingCubeGeom = new THREE.CubeGeometry( 50, 50, 50, 1, 1, 1, materialArray );

MovingCube = new THREE.Mesh( MovingCubeGeom, MovingCubeMat );
MovingCube.position.set(0, 25.1, 0);

scene.add( MovingCube );    

http://threejs.org/docs/#Reference/Materials/Material The key is to set transparent attribute true and set opacity to 0.5(for example).
Add the second the cube which fits inside exactly with no transparency, idea from @WestLangley ( Three.js canvas render and transparency )

backCube = new THREE.Mesh( MovingCubeGeom, new THREE.MeshBasicMaterial( { color: 0xFF0000 }) );
backCube.position.set(0, 25.1, 0);
backCube.scale.set( 0.99, 0.99, 0.99 );
scene.add( backCube );