Transferred my MP3 songs to my Samsung YH-820, but can't hear them?

That's a strange problem indeed.

My first thought is that perhaps the MP3 files themselves are corrupted? Have you tried playing those specific files on your PC and confirmed that they were fine?

Second possibility: are the headphones working properly? Try plugging the headphones into a different device or try plugging in different headphones to your Samsung unit.

Third possibility: is your volume all the way down or do you have a mute feature enabled? On the Apple iPod, for example, you can turn the volume way down, flip the Hold switch, and then be unable to change the volume until you disable the Hold feature. The YH-820 also has a hold switch, on the top of the unit:

Fourth possibility: are you using the Play function properly in the first place? Have you ever had the unit work properly for you? Is there an MP3 demo song that's included with the unit that you can use as a test?

Your view on screen should be like this:

The lower part shows the volume feedback: yours isn't at zero, is it?

The only other possibility I can think of is that either your cable is bad and it's corrupting the files as you transfer them or that your unit is defective somehow. Again, if you've had any success in the past or have certain files on the YH-820 that actually work, that'd be very important information to know.

Hope this set of basic troubleshooting tips is helpful!