How to prevent sliced vegetables/roots from sticking to the blade

You could look at your knife skills. Professional cooks are either faster than ordinary cooks, so the food doesn't have time to adhere to the blade, or maybe their knives are sharper. Do you hone your knife (correctly) before you start cutting?

Another thing is the way professional cooks cut, they use the knife to slice through the food making a slicing motion. Some people just press their knives against the food and apply pressure, which is the wrong way.

Lastly, when dicing, you could make one or two horizontal cuts, then nine or more vertical cuts... without cutting through the base. This way, the food is still attached to the base and will not stick to the knife. Lastly, you turn the food 90º and make the last cuts to create smaller or larger dices.

Edit: There are knife blades designed to reduce the food sticking to them. They have dimples or 'cullets'.

I just ignore slices that stick to the blade. Each typically gets pushed off by the next one, so you only have one or perhaps a couple of slices on the blade at a time. When I've sliced the whole thing, I can wipe them off. This doesn't work when you're rough-chopping, in that delightfully casual way the TV chefs say "just run your knife through it all a few times" since half the stuff is on the blade, but it does work when you're slicing a carrot or garlic or whatever.

Just now, I sliced potatoes thinly for gratin and noticed that I semi-automatically gave each slice a little shove, using one of the fingers from the hand that was holding the potato, just as the slice finished. Obviously this only works for slices large enough to stand up past the top of the blade, but it's a technique you can use for some roots and veggies, even if not for garlic. I'm pretty sure I don't do that for things I intend to chop again in another direction, but only for things I am slicing.

It helps a lot if you wet your knife blade before mincing garlic. It'll still stick to the blade some, but not quite as much, and what's left is a lot easier to brush off.

For bigger things, you can try knives with dimples (like Kyle suggests), and sometimes a wet blade helps here too, but these things only go so far. Stuff is going to stick, and the key is to learn to deal with it sticking. One thing that really helps is to slice with the knife angled slightly, cutting just a bit toward the bulk of the vegetable, so that the slices are tilted away from it and when they get pushed up and off the blade, they fall away from the part you're chopping and don't get in your way. You'll probably still feel the urge to brush it off, but eventually you'll get used to letting it fall.

Beyond that, BaffledCook's tips on knife skills are good - read and upvote them too!

I have this same exact problem and for me the solution was to use a hollow edged knife instead of a hollow ground one. The little pockets on the knife allow the air to separate the slicing easily instead of it being stuck to the blade.

With garlic, give it a rough smash and chop, then let it sit with some salt (preferably coarse) for a couple of minutes. That makes it much less sticky.