Q: How do I keep my run from becoming muddy?
A: Depending on where you are in the country and how your coop is sited, you may occasionally have problems with mud in the run. There are worse things than a little mud from time to time, but if your chickens have no place to escape a mucky run, that means they are pooping in the mud, walking in the muddy poop, foraging in muddy, poopy soup... so ick. That is a formula for illness, and that fertile muck is a perfect place for bad things to grow: coccidia, worms---and worse.
They definitely prefer a green pasture to a mud pit.
When you can't cover the run (that's a huge expense if your run is of any size) and you can't locate in a dry area (you may not have one!), then what do you do? Some use litter in their runs, but we don't recommend it: it's meant to decompose. Unless you are prepared to change the bedding in the run frequently, this is not a good long term solution. Wood chips are far more preferable, as they decompose very slowly. Sand may work nicely as well, but there are some things to consider.
Pros: Sand drains relatively quickly. Plus, it makes for a great dusting area for your birds. And it doesn't decompose, so it doesn't need replacing often.
Cons: it will still require some maintenance--when it's dry, it can be sort of scooped out like cat litter. Plus, your sand bed needs somewhere to drain to, so it will probably need to be slightly raised. You can't simply dig out a pit and fill it with sand (then it will just be a pit filled with wet sand, because everything around it will drain into it.) And sand is heavy, so it can be something of a pain to get into place. And although it won't need replaced as often as pine litter bedding or straw, it will need replenishment from time to time, because your chickens may wallow it out dust bathing in it. Even so, you can still use what you scoop out on a compost pile.
One last thing to keep in mind: sand free of organic matter (like the sand you'll want for a chicken run) is not an especially good medium to grow plants in for foraging, if that's something you're interested in. Then again, if you had muck before, you probably didn't have a lush, green carpet or forage-friendly grass anyway. But as we've discussed before (see the related questions below) keeping your chickens confined permanently to a small run with no access to green places to forage is not ideal. Please be sure to provide them with an alternative. For instance, you can grow a Chicken Salad garden for them.