There is not full agreement among different people. According to the People's New Testament it fell on a Sunday (50 days after Christ rose from the dead) and this seems to make sense to me because the death, burial, resurrection, and blessing of the Sprit all have a meaningful relationship with existing Jewish feats.
Pentecost, one of the three great annual festivals of the Jews, lasted only one day, was on the fiftieth day from the morrow after the passover Sabbath (Le 23:15-22). The Savior, crucified on Friday, was in the tomb on the passover Sabbath, and rose on Sunday, the day from whence the count began. The Sunday following would be the eighth day, and the fiftieth day would fall on Sunday, the first day of the eighth week. Hence, the ancient church observed Pentecost on the first day of the week. For fuller discussion, see Meyer, Howson and Milligan on this passage. All agree that Pentecost come on Sunday. Pentecost was the feast of the grain harvest, and was also held by the Jews to be the anniversary of the giving of the law (Ex 23:16 34:22,23). People's New Testament
I agree with the People's New Testament calculation partly because it matches so well with the symbolism of the events, which seems to be a biblical way of events naturally occurring. Sometimes when we get deep into the nifty gritty about Jewish things, such as this, I prefer to find a source that Jewish Christians support as they are more aware of the Laws of Moses then regular Christians are (although not always of course).
From a Jewish Christian site I found an excellent article on the subject explaining why it fell on a Sunday and why objections to this argument are not well founded:
The fulfillment of these feasts is striking. Jesus died the Friday of Passover week and had to be buried hastily before sunset, which was when the Sabbath began. His body remained in the borrowed sepulchre throughout the Sabbath day, but on that Sunday morning, when the priest was to offer the First Fruits offering in the Temple, Christ arose from the dead, the first fruits of them that slept (I Cor. 15:20).
For forty ensuing days, the Lord appeared to His disciples in His resurrection body, and then ascended into Heaven. Ten days later, the Sunday of the Feast of Pentecost, the Holy Spirit descended upon the believers in Jerusalem and created the ekklesia, the called out body of Christ, the church. These fulfillments were obviously no coincidence, but were part of the overall plan and purpose of God in verifying the powerful meaning of the death and resurrection of Christ, and the establishment of the new body of believers. (Thomas S. McCall)