TheRadio is a web-based radio station aggregator which pulls stations from all over the world into a simple to use interface. You can play music instantly with no fussing or you can build a finely-tuned playlist.
You can search for an artist or genre to act as a seed for your initial playlist or you can go and browse through the channels which cover every imaginable musical combination short of All-Female Eskimo Ska-Tribute Bands. Songs you listen to appear under the history tab and you can easily go back and listen to a previous song again if you wish by clicking on it there.
If you want finer control over future playback you can sign up for a free account and start flagging songs as likes or dislikes by tagging them with a smile or a frown. All of the genre/artist searches you perform are also stored in the favourites tab along with your flagged songs.
TheRadio is a free service and only requires a login if you wish to preserve your settings and favourites between sessions.