Due to a slightly miscalculated water-feature, I have a muddy dining room. The dwarves don't seem overly bothered, but I can't help but feel that it's not doing the engravings justice.
How can I remove the mud from the floor of my dining room?
Make sure there are dwarves with the cleaning labor enabled. They will only clean smoothed, engraved or built floors. Walls will be cleaned automagically if an adjacent floor tile is cleaned, but not otherwise. There is no way to tell dwarves to clean, so it is very much a waiting game.
If they are tracking the mud all over the place you may want to designate the muddy tiles as restricted traffic zones, to minimize its spread (This may be more valuable when dealing with pools of rectum-dissolving demon blood).
Flowing water can displace most contaminants, moving them out of the way, and pumping water purifies it. Admittedly, this is not going to do much to help with mud, which is made by water.
If the tile is made of natural stone, you can also construct a floor over it, getting a clean floor (Thanks to Stuart Pegg for testing).
Also, outside tiles may be cleaned of contaminants on season change (at least it did in earlier version), so you may wish to simply knock the roof through. The added benefit of having a sunlit dining room is that it will prevent cave adaptation (though, if they already have cave adaptation, it will cause them to drop barfs all over the place for a while, which may seem counterintuitive).
If you're not averse to 'cheaty' methods you can use DFHack which can clean the tiles with the cleanmap tool. I'm not sure if dwarfs with the cleaning labor ever clean muddy floors due to muddy floors generally being created by the player in order to farm.