Woolly Mammoth
Woolly Mammoths really should be around today. Isolated pockets of theses creatures lived until historic times, dying out only 4,000 to 6,000 years ago. And we've got frozen specimens with preserved soft tissue. Our great-great-great-great-times-500-grandfathers probably hunted these magnificent beasts and certainly loved to draw them on cave walls. As far as extinct charismatic megafauna go, Woolly Mammoths lead the pack.
Image: Mauricio Antón/PLoS