The origin of this story seems to be a letter sent to the weird news journal Fortean Times in 1999:
FT129: I’ve just learned a truly horrifying thing, and one which I didn’t believe until I tried it for myself. It was told to me by a work colleague, a keen cook, who wasn’t at all amazed when he discovered it.
Perhaps the best way of experiencing the amazement I felt is to try it yourself, in the following manner: take a good-sized piece of fresh liver and place it on your kitchen work-surface. Place a glass of milk 10-20cm (4-8in) away from the liver, on the same surface. Go away and do something else for 45 minutes.
Upon your return you will observe that the liver has crept a good distance towards the milk, leaving a trail of blood! It’s like something out of a horror movie. How, in the name of all that’s decent, does this happen? Fair enough, it may have (in scientific terms) "something to do with enzymes/diffusion/osmosis/whatever", but this is simply avoiding the issue of how the liver "creeps" (something which I am at present far too disturbed to even contemplate observing). I now have an uneasy sense about my internal organs (although perhaps it’s my nervous system, mounting a coup). Ugh.
We’re all colonies of micro-organisms, but it had never occurred to me that some parts might be well nigh autonomous sub-colonies (if that’s the explanation – and it’s the only one I can think of) – my liver, behaving like a slime mould? It’s upset my perceptions of the Universe! If one put beer in place of milk, would the liver crawl away, to avoid liver damage? If placed near a glass of sour milk, would a stomach turn over? What would attract/repel a kidney? Or is this unique to livers? Is this, in fact, the reason they are called livers?!?
My work colleague shrugged off my amazement by saying, simply, "liver likes milk" which is admirably assimilatory of him, but I am as intrigued and bamboozled as I am dumbfounded and appalled. I want an answer, and I want it now. Any answers? In the meantime: milkmen! Do not park your floats outside butchers’ shops…
Garrick Alder, London."
The author appears to be one of the authors of QI, as you can see here. He specialises in quite interesting stuff (apparently, not necessarily scientifically correct, though).
You can also see this 1999 alt.paranormal thread featuring a (quite hilarious) set of puns (including some by noted skeptic P. Z. Meyer)
This thread mentions someone that asserts having heard the myth in 1972.
From a practical standpoint:
- there are no studies that I could find describing this effect
- every single mention I found of this myth was always related to disbelief ("someone told me that..."). Only in one case there was someone reporting the effect happening to them, but they were the proponents of the myth
- from a physical/biological standpoint I see no easy/possible explanation.
In practice, while I cannot provide strong evidence that it's completely false, I also see no evidence of it being true at all, so I shall go with my gut feeling and refuse to believe it.