I'm not sure I can reasonably defend the claim that they themselves were more secular, but the regime used a variety of methods to discipline and constrain religious observance. Especially during the anti-religious campaigns of the Khrushchev era, people in political power were expected to demonstrate their commitment to atheism or at the very least to making religion (particularly Islam) a small part of an otherwise modern, Sovietized lifestyle. Anything that hinted at religious "extremism" (for example, advocating the veiling of women) was marginalized through censorship and could have you put in prison if you were associated with it.

Lots of people continued to attend religious services (especially Orthodox Christians, who had a privileged status vis-à-vis the state, but Muslims as well). It's quite possible that their serious exposure to religious teachings was relatively minimal unless they had some sort of special reason to seek it out. By the same token, though, more extreme forms of Islam and policitized Christianity became a lot more appealing in the 1980s as Soviet society began to unravel--they could plainly offer something mainstream socialist ideology could not. Also, the war in Afghanistan led to a lot of Soviet citizens becoming radicalized in an Islamic direction, since it explicitly pitted the two against each other in a military conflict, potentially shattering the mainstream notion that a disciplined, culturally impotent Islam could coexist harmoniously with Soviet socialism.

For more, check out Adeeb Khalid, Islam After Communism, and Douglas Northrop, Veiled Empire (both of these are about Central Asia rather than the Caucasus, but they're a good start).