Indirectly, there is some truth. Prior to the fall of the Soviet Union there was a conscious policy to woo muslims, we managed to pull Egypt out of the Soviet camp, for example, and we worked hard to try and draw Syria and Iraq away from them though not very successfully.
And we had a dream of using muslims to dismantle the Soviet Union, the idea was that if we could ever provoke a muslim insurrection the USSR would come apart at the seams. It all made perfect sense when we were at war with them.
This led to a big error, when we allowed the Shah of Iran to be replaced by the Ayatollah, believing that the Islamists would be more ruthlessly anti-communist. And it had one big success when we, with the Saudis and the Pakistanis, put together the Afghan insurgency that led to the Soviet defeat there.
With the implosion of the Soviet Union, it became possible to implement the policy when it no longer made any strategic sense. The Turks had their dream of uniting the Turkic countries of Central Asia, and the Saudis had their dream of building their Wahab empire, and since we were allies with both of them, we didn't oppose either of them.
Consequently, under Clinton, we steadfastly refused to say anything against the Chechens during their first revolt. Popular opinion began to shift after they won defacto independence and rather than build a real country, allowed their territory to be used for further war against the surrounding territories. But while Clinton was still in office, our official policy didn't change.
When Bush took office, his first foreign policy step was to embrace Putin, and that signaled the beginning of our pullback from support for the Chechens. 9/11 sealed it, and after 9/11 we declared war on every one of the Saudi insurrections across Asia and gave complete approval to Putin to do whatever he had to do in Chechnya. The only Saudi insurrection we did not attack was their outpost in the Balkans. We have sought to offload that to the Europeans, though.
It should be clear, though, that our "support" for the Chechens was rhetorical. They were a Turkish and Saudi operation, mostly Saudi after a time. And our patience for Saudi foolishness began to end with GW Bush's administration, and ended completely with 9/11 and their reluctance to help us either in Afghanistan or Iraq either one.
Including Bosnians.
Russia's response to the ongoing criticism was to put together and take on a dogandpony show a compilation of videos the chechens had made of themselves, torturing and mutilating children and other hostages.
I recall reading the responses in several countries to these showings. I believe it was in Helsinki, where a virtual nest of pro-chechen supporters and activists thrives, that a woman passed out and several left the room to vomit, after seeing only part of the video.
It was a most effective vehicle for showing the true nature of the chechens, who raise children to endure in a brutal warrior-adoring culture at best. In most of the videos I have seen, the chechens gleefully hold up body parts still dripping with blood from their victim and the smiles are just not faked.