Did Adam and Eve ever practice animal sacrifice? Genesis 8:20 (NIV) Then Noah built an altar to the Lord and, taking
  some of all the clean animals and clean birds, he sacrificed burnt
  offerings on it.


Noah sacrificed burnt offerings. What about Adam and Eve? Did they ever perform animal sacrifice?
Premise

We cannot conclusively say that Adam and Eve performed the any sacrifices, but the evidence is very heavy that they did and it was ordained by God and likely that He instructed them in it or even did the first one Himself.



It is definitely clear that sacrifice was instituted very early.

Abel and Cain apparently performed sacrifices, Abel bringing animal material and Cain bringing plant material. The ensuing rivalry between the two led to the infamous 'first murder'. It has been argued that Cain knew animal material and animal life was the requirement and his obvious 'cannot be wrong' attitude and jealousy was what led him to murder his own brother. 

But did Adam and Eve sacrifice?

One should think that Cain and Abel were told by someone that it was necessary. It is possible that God Himself told them to do so, considering they were also on the same plain speaking terms, demonstrated when God questions and subsequently curses Cain. However, Genesis 3:21 is curious stating just after God declared punishments for the fall:


  The Lord God made garments of skin for Adam and his wife and clothed them.


God Himself made the garments for Adam and Eve. Depending on your views of whether there was death before the fall this may or may not be significant. Since you do not specify a tradition I will stick to literal Bible interpretation, in which this view, that there was no death mentioned until this verse, therefore, no death before the fall, is well supported.

So many have hypothesized that God Himself actually performed the very first sacrifice for Adam's and Eve's sins. This site argues that for example. Always leading by example, Adam and Eve then mimicked God's righteous actions and began performing sacrifice. Whether you take this view or not it is indisputable that God approved of the animal's killing so that would leave only Adam or Eve to actually perform the sacrifice. It really does not matter, though, who did it first because just one generation later Cain and Able are sacrificing as if it is a regular thing, showing that it was already instituted and the most likely time was Genesis 3:21. Also, if God did this first sacrifice, He did not do any others, again leaving the task to Adam and Eve.

Now it is not until the reveling of the Law that frequency and reasons for sacrifice were known to us at least. This is more than 1000 years later. It is clear that there were thanksgiving reasons, as Noah did in your example, and it is implied that Cain and Able were performing a possibly annual or other time frame sacrifice as required by God. There are also a few examples of reasons for sacrifice in the story of Abraham. The most notable being told to do so by an angel of God when Abraham was told to sacrifice his own son Isaac. So there may have been very well understood reasons by the ancient persons for sacrifice that may have even been started with Adam and Eve shortly after the fall, but the only indication of those 'rules' becoming written is Leviticus with Moses. It is entirely possible that those sections in Leviticus were only parroting what had been done since the beginning.

Conclusion

So we cannot conclusively say that Adam and Eve started the sacrifice rituals, but the evidence is very heavy that they did and it was ordained by God and likely that He instructed them in it or even did the first one Himself.