If sin blocks prayer, and if we're all sinners (at all times, right?) then how do anyone's prayers get answered?
Most of us tend to live on the Solzhenitsyn line, going back and forth between good and evil. The difference is in trying to self examine and judge ourselves, and thus repent. In this way, when we meet the true and just Judge, we will have already considered such things, and we can have hope in God:
"For we have not a high priest who is unable to sympathize with our weaknesses, but one who in every respect has been tempted as we are, yet without sin."
Our prayers, if rightly directed, give thanks and glory to god for all things. In this manner, they are offered up to God as a remembrance, a sacrifice in and of themselves, as are our good works, as Cornelius prayers and acts were in Acts 10.
Sin distracts us from prayer.