You need to care about the texture as that is how you color the object. Edit the texture for the diffuse color and apply a gradient. Then create a new material, this will save that gradient as a material, you can then apply that gradient to the other sides. Finally edit the sides that don't match to your liking.
As for true 3d rectangle you can use the pen tool to create your rectangle, then extrude it.
I made 1 mistake, if you apply the material to the remaining sides, then all sides share the same material and therefore the same texture file. So, editing that texture would update all sides. Therefore any sides that do not match need there own file. So the sides could share and the top and bottom have a different texture file.
Sorry about that...
Let me be candid: Your unwillingness to adapt to the specifics of 3D is your undoing. If you "don't care about the texture" you have already lost. Things simply don't work that way. Just the opposite is true: You have to go in and edit the texture and place the same gradient to each side based on the unwrapped UV projection *hint, hint*.
I do appreciate the ideas here......
FIRST - I don't care about the texture in the sense that I'm not trying to give the 3D object a special type of texture or roughness, like marble, cloth, ridges, knurl marks.................... hey, if the only way to color the sides of a 3D object is through the "texture", well gee great - tell me how to color the sides. I just want it colored with a gradient and to keep it smooth, NO special textured appearance.
An interesting realization here. The gradient color I want here on my 3D object is an attempt to represent lighting. Perhaps this "faking" the lighting brings about a situation that should be solved with the 3D light tools?
I WAS able to use the Cube Warp mesh - to create a true 3D rectangle, but when I edit the texture for diffuse color - it applies the gradient color in a strange way. It applies the gradient on the 3D rectangle face, and top separately, rather than "wrap" the gradient starting with the face of the 3D rectangle. You can see this in the image below:
So.... is there a way to "wrap" the gradient, rather than apply it to the face and top independently?
Also.... my diffuse texture is a blue and white gradient...... why did it come out blue and gray?
The effect with the gradient that I am trying to achieve, is shown below. You can see where the white center of the gradient on the face flows across the top as if it were the flow of the light. I hope this helps show what I'm trying to achieve. I can use the 3D mesh - cube warp to get the 3D rectange, I just don't know to color it with a gradient color like this?????