I've been cringing at the suggestions of cutting, shaving or even covering over roots. Barring mechanical intervention or unintended erosion, tree roots appear where the tree wants and needs them to be. Removing them, altering them or covering them risks damage or even death to the tree. The consequences may not appear immediately but show up over years until its too late. Damage to a tree's root system can put the tree into stress thus making it an easier target for disease or pests.
IMHO The best suggestion I can see is the raised walkway built in the fashion of a deck.
Not sure if anyone is still reading this or still looking for advice. We have an oak tree that is about 100' tall and the center of the trunk is about 5' from our sidewalk, and 10' from our front porch. The area between the sidewalk and porch is a flower bed.
The tree roots have pushed parts of the sidewalk up and over, so it's a tripping hazard. After having a certified harborist come out (with a PhD no less), we were told that we could shave down the surface roots - no more than 6" down - without harming our tree. It's just going to be one major root that is approximately 1' thick, if not more. We are going to jack up the concrete sidewalk, get them to come out and shave that root down, then rebuild the sidewalk further from the tree and closer to the porch. They've already come out and shaved down roots from our side yard (next to the driveway) but that area was well outside of the drip line. With our tree, it's so old, mature, and stable, that grinding down a few roots 6" wasn't going to do any harm to the tree.
Our fix won't solve the problem of roots growing under the concrete sidewalk, but it did save the tree. The other options were to cut the tree down, trench 24" down around the front porch (which would have created structural problems and with high winds that we get from hurricanes and storms, the tree could possibly topple onto our house), or build a deck over our existing porch and sidewalk that would allow the roots to grow under.
So basically I'd call in a local certified arborist and discuss your situation with them. Having roots cause problems with walkways is nothing new, and they'll be able to give you sound advice on what actions you can or shouldn't take.