One major premise of dispensationalist theology is that the promises made by God are to Israel and Israel only. The church (NT) is thus a "parenthesis" between the reign of Israel. A general description and links to sources is available here.
So with that setup, I'd like to go after that one assumption, that God's promises are to Israel and Israel only. Now, this point doesn't establish that the church is the new Israel, but it certainly does demonstrate that such a view certainly wouldn't be inconsistent with God and His revelation.
In Matthew 3, John the Baptist is baptizing people in the wilderness and he sees the Pharisees and has these loving and kind words to say to them -
9And do not presume to say to yourselves, 'We have Abraham as our father,' for I tell you, God is able from these stones to raise up children for Abraham. 10Even now the axe is laid to the root of the trees.What does this mean regarding dispensationalism?
Simply that those promises that were made to Israel could (and I would argue are) be fulfilled by a body of Gentiles, even though they are not ethnically Israel, because they would be children for Abraham raised not out of stones, but out of the pagan nations. They are grafted in (Romans 11:17-24), children of Abraham by faith (Galatians 3:7)