It is not wrong to try to break groups of things apart and understand them each individually. However, they must always be put back together to understand what is being said.
The Meaning of the Total Phrase
Initially, the NLT's translation is appealing:
They are reborn—not with a physical birth resulting from human passion or plan, but a birth that comes from God.
But this translation robs the statement of profundity by placing the onus on the nonphysicality of the birth. In broad context, the biblical theology of the σάρξ/πνεῦμα distinction does not mean primarily physical/nonphysical but man/God. In the specific verse, it seems that the statement is stronger than that.
Regeneration is a major theme in John, and we see it here as early as chapter 1. The three things together indicate that the rebirth effected by the Spirit of God excludes any sort of human cooperation that you could possibly think of. The number three indicates completeness of reference. For this reason, determining the exact reference of each is less important.
The Meaning of αἱμάτων
Bob Jones suggests the bloody act of circumcision in his answer; Gill mentions it too. To explore this idea further, I have asked the question, Does a birth-circumcision connection exist in the Old Testament? If not, I find this connection here to be much less likely.
The bigger objection, though, is that this does not account for why it the word is in the plural. Calvin takes it to indicate heritage:
The reason why he uses the word blood in the plural number appears to have been, that he might express more fully a long succession of lineage; for this was a part of the boasting among the Jews, that they could trace their descent, by an uninterrupted line, upwards to the patriarchs.
This explanation also makes more sense than taking bloods to mean physical. For all I can see, that is just farfetched. Ancestry is the clearest reading.
The Meaning of θελήματος σαρκὸς and θελήματος ἀνδρὸς
Calvin takes both these phrases to be equivalent, and this certainly is a viable interpretation which does no violence to the meaning of the overall phrase. However it seems to me that the details of each phrase may be determined more even if they are taken together as a hendiadys referring to human will. (Gill breaks it down a little more than Calvin.)
Given that σάρξ often denotes that which is human (and its precise meaning depends on the context and what it is standing in contrast to), and is a mass noun, I might paraphrase thus: not by any human power of the will nor by any man's decision because it seems to me that the will of flesh refers to the human capability of decision and will of man refers to a specific individual's decision.
See also Henry, who groups the last two phrases together in much the same way as Calvin.
[I am planning on buying some commentaries on John. Perhaps I will post more then...]
Of the three things mentioned in the answer, only one (Jewish heritage) is being referred to by this verse. The other part of the verse refers to the human capability of act of decision.