I'm pretty sure I've only heard two ways that artists deal with clearing samples in music. They either go through the process of contacting the labels or estates or rights holders to ask permission, usually because of a label fearing a lawsuit, or they fly by the seat of their pants, ignore any legal process, and release the music with uncleared samples.
Legitmix is a new service that allows artists to upload their mixes, and then tag each sample they used. The way the sample is cleared is that when the source track is tagged, the purchaser of the remix must also purchase the source material, unless they already own it. In which case, Legitmix's software will automatically add the samples in.
It's an interesting concept, but it can potentially double the cost of a song for the listener once the source track is bundled in. Instead of setting up a royalty scheme where a percentage of a 99 cent download goes to the rights holder, the listener foots the bill up front for any source material they don't already own, which I am imagine in some cases could contain three or four tracks and bring the price for a digital download of one song close to $5.
Check out a demo of the service from Billy Van:
UPDATE: Legitmix has also posted this video, which explains that the file a listener purchases doesn't contain any samples, but instead allows the listener to recreate a song using source material they already own, or provides the option of purchasing any tracks they do not.