It's hard to see why EMI are suing sideload.com - it's just a search engine for music files that exist all over the web; nothing's hosted locally.
It's a bit less hard to see why they're suing mp3tunes.com, because from a technical point of view it makes a copy of music files - either from the web (which might not be public domain, but are still freely available) or uploaded from your own collection (which you're presumably allowed to make a copy of for personal use) - and then lets you (and only you*) listen to them later. The last part is where the trouble might be, as their server is making a copy of the file for you rather than the other way round.
What they're complaining about is that MP3tunes has built "a commercial, for-profit business dependent on providing their users with access to popular copyrighted music" without authorisation from each record label. Which sounds like a great business to me, but apparently not to EMI. A random website may be authorised by EMI to make available an MP3 file of a song owned by EMI, but no-one else (in EMI's view) is allowed to link to, copy or redistribute that file, especially not for profit.
It's a ridiculous complaint, but either way, this copyright situation needs fixing so it doesn't keep happening. If this case stands up in court, then any site that lets you make copies for your own use of mostly copyrighted material could be affected, including CiteULike for example (PDFs).
It's interesting to note that this situation arises only where digital files are made available behind payment barriers (PDFs, MP3s) or at URLs that are unlikely to remain fixed. In contrast, for sites like Flickr - where the creator pays to have their content stored and put online - people are quite content to (hot)link to the files at their canonical URL rather than make a separate copy elsewhere**.
Similar to the decision that Google shouldn't be prevented from making a copy of someone else's digital files in order to build a search index, I don't think it's right that you (or someone else acting on your behalf) should be prevented from making a copy of a digital file that you own - or that is freely available on the internet - for your personal use.
Details at Michael Robertson's blog, including the legal documents.
* I don't know how you could stop people sharing the password for their MP3tunes account, but that should be their individual resonsibility, as long as the host responds to legitimate complaints of copyright infringement in the same manner required of any public file hosting service.
** Not quite a fair comparison, because Flickr has everything on one site, and people are less likely to want to build an external collection of other people's images.