I believe the reasoning is that Christ was ressurected on Sunday, so the observance of the Sabbath was shifted to that day as a celebration of the ressurection.
As far as I know, there is nothing in the Bible that says to "keep sunday holy" - other than an idication in Acts 20:7 that the disciples met on Sunday to "Break Bread" (the Lord's Supper).
Now, if you are willing to accept modern revelation (which I am), the waters get less murky.
http://scriptures.lds.org/en/dc/59 (Lord's Day = Sunday, from the reasoning above)
9 And that thou mayest more fully keep thyself
unspotted from the world, thou shalt go to the house of
prayer and offer up thy
sacraments upon my
holy day; 10 For verily this is a
day appointed unto you to rest from your labors, and to pay thy devotions unto the Most High;
11 Nevertheless thy
vows shall be offered up in righteousness on all days and at all times;
12 But remember that on this, the
Lords day, thou shalt offer thine
oblations and thy sacraments unto the Most High,
confessing thy sins unto thy brethren, and before the Lord.
13 And on this day thou shalt do none other thing, only let thy food be prepared with singleness of heart that thy
fasting may be perfect, or, in other words, that thy
joy may be full.