Brothers and sisters: Since we are surrounded by so great a cloud of witnesses, let us rid ourselves of every burden and sin that clings to us....
The natural thinking is that this "cloud of witnesses" comprises the Christian saints in heaven.
But the pericope cuts out the first word of the verse, which is, "Therefore." I guess you can't very easily start a reading with, "Therefore," since it suggests you're boarding a train of thought in the middle.
In this case, Hebrews 12:1 is the middle of a train that begins back at Hebrews 11:1, the first word of which is, "Faith." Faith, in particular the faith of "the ancients" is the key to all of Chapter 11. Abel, Enoch, Noah, Abraham -- Abraham above all -- Isaac, Jacob, Joseph, Moses, the Hebrews who followed him out of Egypt, Rahab: all are mentioned, and then in verse 32:
What more shall I say? I have not time to tell of Gideon, Barak, Samson, Jephthah, of David and Samuel and the prophets....
If we put the end of Chapter 11 together with the beginning of Chapter 12, we can make sense of that "Therefore":
Yet all these, though approved because of their faith, did not receive what had been promised. God had foreseen something better for us, so that without us they should not be made perfect.
Therefore, since we are surrounded by so great a cloud of witnesses, let us rid ourselves of every burden and sin that clings to us and persevere in running the race that lies before us while keeping our eyes fixed on Jesus, the leader and perfecter of faith.
The "cloud of witnesses," then, refers to those who came before Christ, from Abel all the way down to St. John the Baptist (not named, but surely to be counted among the prophets who "went about in skins" and "wandered about in deserts").
What they give witness to is not the death and resurrection of Jesus for the salvation of the world, but, more simply, faith in God. An imperfect faith in God is, of course, a lesser thing than a perfect faith in God's Son, but for that very reason we who have received what had been promised must not be found less faithful than those who did not. Therefore, let us listen to their testimony and persevere.
Does this mean the expression "cloud of witnesses" doesn't also refer to the saints who came after Christ? Not at all. (I know; I had you worried there for a minute.) Their witness is no less relevant to those of us who have not yet finished running the race that lies before us.
But we should not forget the witnesses who came before Christ. They are as much members of the Church Triumphant as St. Peter or St. Francis of Assisi.