I see two problems with this sentence. The first is the absence of 'the' before 'precious things'. The second is the noun that the defining clause refers to. It is illogical to refer to 'diamonds' that were on the wall. To avoid any ambiguity, I'd paraphrase it as follows:
- They stole the precious paintings that were hanging on the walls as well as the diamonds.
- Not only did they steal the diamonds, but they also stole the precious paintings that were hanging on the walls.
Do you mean 'The diamonds, as well as the precious paintings that were hanging on the walls, were stolen'? I think this usage of commas could mean that those precious paintings are not so important to you as the diamonds. I prefer, "The precious paintings, which were hanging on the walls, as well as the diamonds were stolen."