Anything made by the Red Hot Chili Peppers in the last 10 years. Some people didn't like their new album, but I found it to be an album I could listen to every single song from start to finish. Your mileage may vary.
White Stripes are pretty good, at least from what I've heard of them. Good enough to give them a try, anyways. A lot of people seem to like Jack Johnson, but I never really got into him so I can't tell for sure.
Within the last 10 years...?
-Enigma (particularly "The Screen Behind the Mirror", "Voyageur", "Le Roi Est Mort...", and "A Posteriori")
-Amethystium (anything, although if you're wary then just peep "Emblem")
-Ziggy Marley "Dragonfly"
although lately i've been mostly listening to Smashing Pumpkins' "Mellon Collie..."
I know you said no rap, but you owe it to yourself to find Aceyalone's "A Book of Human Langauge" if you don't already have a copy...