While Robert Thieme was a dispensationalist and a Free Grace theologian, thus getting so many things right, this is an issue where I heavily disagree with him.
A Theological novelty
As far as I am aware, there is no concrete proof that anyone before Thieme held to the view that the actual blood of Christ was not a part of His atoning work on the cross. In fact, John Walvoord also says that he knew of no one who agreed with Robert Thieme, as Walvoord wrote this letter to Thieme in 1972:
Thank you for your letter of September 24 and your previous phone call relative to the booklet, "The Blood of Christ" by Robert B. Thieme. Although Mr. Thieme quotes a syllabus of mine written many years ago, I did not then nor do I now agree with the point of view expressed in this booklet . . . . I do not know of anyone who follows the point of view of Mr. Thieme, and his teaching on this point has apparently confused a number of people. It is not the teaching of any member of the faculty of Dallas Theological Seminary 1
Walvoord continues:
While He did not bleed to death, I do believe that He literally shed His blood as part of the act of dying and that this was necessary to fulfill such scriptures as Hebrews 9:22; 1 Peter: 18-19; and similar passages. The implication from your quotation of my writings that you and I agree on your major thesis set forth in the booklet is, therefore, wrong as I do not agree with the major point of view which you express in the booklet. Further, I do not know any Dallas faculty member who shares your point of view. In fact, your point of view is quite a unique and unusual interpretation 1
Biblical references
The necessity of literal blood in atonement becomes apparent through the comparisons the New Testament makes to the Old Testament. Take the Passover, for instance. During the Exodus, the lamb could not just merely be slain any way and left it at that. The literal blood had to be applied on the door (Exodus 12:13). Jesus as our passover lamb (1 Corinthians 5:7) had to also shed His blood for God's wrath to be satisfied.
Similarly, in the Old Testament, if you killed an animal in any other way than shedding its blood, they would not be counted as valid. The book of Leviticus is explicit in its affirmation that the blood had to be sprinkled (Leviticus 16:19), the book of Hebrews follows this theme, saying that we are saved through the sprinkling of the blood of Christ (Hebrews 10:22) and that we have boldness to enter the Most Holy Place because of the blood of Jesus (Hebrews 10:19).
Consider also Satan's temptation of Jesus in Matthew 4. If any way of death could atone for the sins of the world, why would Satan tempt Jesus to jump fall off a cliff? Jesus's choice of death on the cross underscores the importance of blood sacrifice in atonement.
References
1:Bob Thieme's teachings on Christian living By Joe Layton Wall