Is there anywhere to get free testnet bitcoins?

I'm building a project with bitcoins and want to test it with testnet. However, I don't have any testnet bitcoins, and despite the difficulty being less than 100 usually, I still haven't been able to mine a block on my PC (CPU only) after a day. Is there anywhere I could get just some small amount of testnet bitcoins for testing?

(Link to faucet URL removed... please see the community wiki answer further down the page for links of faucets.)

The term "faucet" has been used for quite some time for a site giving away small quantities of coins for free. If all links in this page stop working, a web search for "bitcoin testnet faucet" may find a different site.