Well, that's what I did. Here's the before and after:
I put a small white slimbrick on the larger one so that when I took it off you could see what hadn't been cleaned. However, I didn't reckon with a rather surprising feature of washing Lego bricks in a washing machine while tied up in an old pillow case. Here's what the above piece looked like when I took it out of the pillow case:
When I put it in, it had one brick on it. I was actually worried that this might fall off. However, when I took it out, it had another six.
This happened a lot. I separated every brick before I put it into the pillow case, the only exception being the small brick I attached to the larger one for the before and after photo. Many, many bricks joined themselves together in the washing machine. Here's another example:
This base plate brick, which is the same size as the other one, managed to attract thirteen bricks to it. Some of them are on very tightly, too — they're a real challenge to break apart. What's even more weird is that I didn't put the wash on spin, they just got tumbled at a low temperature on a quick wash followed by a drain.
Although I'm generally pleased by the results, there are some bricks that didn't come out too well. Here's an example:
I'm not sure whether it's the detergent I used (hmm, maybe I shouldn't have used detergent?) or the hard water. It doesn't look like scratching. I'm now experimenting on individual bricks to see what gets rid of it; I'm thinking that leaving it to soak might do it.
Somewhere, some artist is right now throwing an assortment of Lego bricks into a washing machine with the intention of selling the art it creates on Etsy...