Education about injustice, promoting unity, eradicating bad language, self-esteem, equal-protection steps, etc. are nice, but superficial. However, when a person of any ethnicity is deeply offended, basic human nature tends to lead him/her to forget all these nice things. So the problem of bigotry, etc., remains.
Since so-called "racial tensions"* matter is a moral issue, I think much of the answer lies within the Church (no, this is not about religion, but relationships). If man's heart is in the right relationship with his Creator, chances are his relationships with others made in God's image will be healthier. His loves flow through us, and we love others.
When Christians remain dormant, this gives opportunity for haters to rise up and "placate" hurting people and entice the ignorant.
*BTW, the tensions can be applied between any group of individuals, not just on race, because there's only one race- human.