What is the word for working hard to "take the concerns of someone away"?

Solution 1:

You put their minds at rest.

This is a common idiom to suggest that you will prove to them that they need not exercise their minds with worry.

Solution 2:

Assuage

verb (used with object),

assuaged, assuaging.

to make milder or less severe; relieve; ease; mitigate: "to assuage one's grief; to assuage one's pain."

to appease; satisfy; allay; relieve: "to assuage one's hunger."

to soothe, calm, or mollify: "to assuage his fears; to assuage her anger."

Via dictionary.com Source

Solution 3:

If you want to eliminate their concerns rather than merely make them less concerning then I would suggest that you work to Dispel their concerns.

Merriam Webster:

to drive away by or as if by scattering

Example sentences, also from Merriam Webster:

This report should dispel any doubts you have about the plan.

She made an official statement to dispel any rumors about her retirement.

The experience dispelled some of our fears about the process.