To download free music on android device, you must have an app. MP3 Music Download allows you to download free MP3 music to your Android phone. Other apps that download free music are I Music, Music Search and Easy Mp3 Downloader for Android.
Something I've noticed about last.fm is that they play too many of the same songs over and over again (kept on noticing the message "you've listened to this artist once before, and this song once before", that kind of thing.) Pandora is a good app but right now I'm not sure if it would have that same problem. So I understand you asked about downloading but streaming music is usually a better way to go because that's free or much cheaper (depending on the service) and its LEGAL. Don't assume you'd never get caught, there have been people who have paid some big fines over a few free songs.
Someone on here recommended MP3Skull - they are right, you could download mp3s for free there but that's not to say its legal to do so. maybe do it from your laptop when using public wifi so its not traced backed to you?
Fast Download MP3 Application for Android. Search mp3 music you want and download them to your mobile. Easily search and find all songs, download mp3 music and refresh your music library, listen with music directory playlist. Also you can listen mp3 music files with our built-in player. This application can help you get best results from public mp3 sites with mp3 downloaded specification.