I don't exactly understand how would ray-tracing change anything, here is how it usually goes:
You can have a texture for any parameter you like, but the most common is a diffuse(color) map, so you just sample this based on the texture coordinates and use the texture in place of the single-color diffuse. You could also combine the texture with an arbitrary color, tinting the given element.
Other common textures include normal maps, reflection maps, which replace their respective counterparts. Normal maps can be a bit tricky, because they specify the rotation relative to the surface, so you need to construct a Tangent Bitangent Normal matrix, which will do the transformation for you.