Java 8 may be used for 3D work.
You can download a
Java 8 preview version.
3D features in Java 8 are not officially documented yet, but there is a good overview with usage notes in the open-jfx wiki:
https://wiki.openjdk.java.net/display/OpenJFX/3D+Features
To display a rectangle with an image on it, you could use a MeshView with a texture set on the Material attached to the MeshView.
http://download.java.net/jdk8/jfxdocs/javafx/scene/shape/MeshView.html
http://download.java.net/jdk8/jfxdocs/javafx/scene/paint/PhongMaterial.html#setDiffuseMap%28javafx.scene.image.Image%29
Java 8 also includes a Box which you could use to model your cube, but using different materials (texture images in your case) for different sides, might be simpler with multiple MeshViews.
http://download.java.net/jdk8/jfxdocs/javafx/scene/shape/Box.html
Here is sample code for using a MeshView in Java 8 (it is early access, so you may need to make some modifications for it to work with the latest Java 8 version):
https://wiki.openjdk.java.net/display/OpenJFX/SimpleMeshView.java
The sample program provides high level settings for the MeshView within the JavaFX scene graph.
Animating portions of a 3D scene graph can use standard
JavaFX transitions.
JavaFX is handling the transformations for the SceneGraph and offloading some of the MeshView geometry math and rendering to the GPU via internal OpenGL or DirectX implementations.
If your 3D model is defined in a standard 3D modeling format, such as .obj, you can directly import it into Java 8's 3D system using the Interactive Mesh 3D model importer:
http://www.interactivemesh.org/models/jfx3dimporter.html
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You can do 3D in JavaFX 2.2 (Java 7), but the API is a bit clunky, largely incomplete and pretty buggy.
A JavaFX 2 implementation of a cube with mapped images on its sides:
http://learnjavafx.typepad.com/weblog/2011/07/earthcubefx-on-javafx-20-download-from-osconjava-2011-keynote.html
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