This was definitely a reason I got my iPhone in the first-place, but it took me a few months to gather my wits and actually endeavour to do this.
First, you're going to have to jailbreak your iPhone if you want to do a direct download. Technically, you can just Safari-browse to an OCremix's webpage and play it streaming through the browser and not jailbreak the phone, but that's a waste of bandwidth if you do it more than once, plus cell phone signal isn't always great at (*insert job location here*).
So, if you've decided to jailbreak the phone (which, if you're reading this far, have enough interest to possibly do it), go ahead and get that out of the way. There's plenty of web resources for how to do it, so go ahead and get to googling. If you end up having trouble, I'll try to find you some of the resources I used.
Next, make sure you have Cydia installed. You're going to want to install the following:
1. Safari Download Plug-In
2. dTunes
The download plugin allows any MP3 you try to access in Safari to be downloaded directly to an iPhone local directory. dTunes is a media player that plays MP3s directly from said download directory, and features editing and browsing so you can organize the musics you get.
Hope this helps!
Edit: To address Taucer's "just download" thing: the iPhone can stream stuff from the data connection, but doesn't let you download anything like that. You have to be connected to WiFi to buy and download anything on the iPhone's iTunes store, and furthermore any games over 10 mb need a WiFi connection to download. To get new stuff in the library, most people have to download it on their computers and sync with iTunes, which is a PAIN more often than not. Jailbroken iPhones have more capabilities for doing things on-the-fly than the vanillas do.