As of Wednesday, January 23rd, you can now play full-length tracks and entire albums for free on the Last.fm website.
From the Last.HQ blog:
Something we’ve wanted for years—for people who visit Last.fm to be able to play any track for free—is now possible. With the support of the folks behind EMI, Sony BMG, Universal, and Warner—and the artists they work with—plus thousands of independent artists and labels, we’ve made the biggest legal collection of music available to play online for free, the way we believe it should be.