Relations may not be all warm & fuzzy, but neither are our relations with China. America's biggest trading partner.
If we really hated you, we would be applying economic sanctions, and a great many of our politicians would be beating the war drums; a la Iran.
Yes their is the whole Missile-Poland affair, but these missiles are most likely meant to stop Iranian/Pakistani bomb threats than Russian threats.
In term of media coverage, Iran gets all of the (bad) press, Russia is rarely mentioned. At least with the papers I read anyways.
Possibly you don't read all the articles in Western papers which defend Chechen terrorists but they get quite some notoriety in Russia for the obvious reasons.
I'm was very young when the Chechen Wars took place and so can not remember how the West viewed them.
However from what I read on AGC, most Westerners (especially Americans)view the ongoing Chechen Islamist Insurgency in a very negative light. As they do with any Islamic political movement.