The Hebrew Bible is a treasure trove of truth, and provides the lens through which to understand this passage regarding the Greeks seeking Jesus.
First, the triumphal entry into Jerusalem resonated not with the Feast of Passover & Unleavened Bread (springtime), but the Feast of Tabernacles (autumn). That is, when the people took boughs and palm branches to welcome Jesus, they were singing from Psalm 118, which was the psalm describing the surrounding nations (Ps 118:10-14), and finally the cry that the Lord save from these nations (Ps 118:25). In other words, "Hosanna" was meant to convey: O, save us (from the Romans because you are the King of Israel).
The second imagery of Tabernacles hearkens to the wilderness journey of 40 years. That is, the people, notwithstanding that they were geographically located in the land promised to Abraham, were "not at home" in their own land because of the subjugation by the Romans. To reinforce this point, the triumphal entry into Jerusalem by Jesus hearkened not to Solomon entering Jerusalem (1 Ki 1:38-39), but to King David mounted on a foal who entered Jerusalem from the Mount of Olives. (Please click here for more detailed explanation of the parallel.) The point here is that in direct parallel to King David, Jesus also mounted on a foal from the Mount of Olives and entered Jerusalem, which (parallel to the situation and time of King David) was dominated by an illegitimate power on the throne. While the argument that the parallel could apply to Solomon (since Adonijah usurped the throne of David at that time), the question of "Why two animals instead of one?" is only solved if we compare the triumphal entry of Jesus to David.
Now to the question.
The Greeks that now appear on the scene take us into the Feast of Unleavened Bread, for which they had arrived into Jerusalem (John 12:20). These Greeks seek Jesus in the same way some of the disciples found Jesus; that is, by the concatenated word-of-mouth process of Andrew and Philip talking to others who became disciples of Jesus, which takes us back to John 1:35-47. But Philip and Andrew received the calling directly from Jesus, with no intermediary (as was not the case with Simon Peter and Nathaniel, who were called to Jesus through Andrew and Philip, respectively). Philip and Andrew now appear to Jesus, who indicates that he (Jesus) himself will draw all men to himself (John 12:32, which is compared to John 6:44 to indicate how this direct calling will occur).
Jesus will therefore draw men to himself by "dying like the grain of wheat" (Jn 12:24), and then being "lifted up." These Greeks came to Jerusalem to celebrate the Feast of Passover & Unleavened Bread, which commemorated the dying of the Paschal Lamb, and the subsequent release from the bondage of Egypt. The Feast of Passover & Unleavened Bread therefore was about those who killed the first-born lamb (and thus suffering "lost" life and then escaping Egypt) versus those who remained in Egypt and "lost" life, when the Angel of the Lord killed the first-born sons of Egypt. The parting of the Red Sea and the resurrection of Jesus are also in direct parallel in the Feast of Passover & Unleavened Bread, when the armies of Egypt were destroyed as was the power of Satan: thus Jesus adds, "...the ruler of this world will be cast out" (John 12:31). Please click here for a detailed analysis of those parallels.
Finally, the way that Jesus will draw all men to himself is by being "lifted up" (Jn 12:32) in the same way that the as the bronze serpent was lifted up in the wilderness (Jn 3:14), which brings us back to the imagery of Tabernacles. While these people escaped Egypt, they were only saved from the sting of the serpent of sin (whose infection was death) by believing the word of the Lord and looking upon the bronze serpent "lifted up."
The point here is that if you connect the dots and believe the word of the Lord, you see the light (Jn 12:35-36), since the question of being "lifted up" contradicted the then current paradigm of the Christ as never dying (Jn 12:34). This explanation of Jesus was rejected by his listeners (Jn 12:37) notwithstanding that few had believed (Jn 12:42). The words of Isaiah cited concerning the "blinding of the eyes" of those who see and especially the "hardness of heart" (Jn 12:40) hearken again back to the Exodus of Egypt, when the Israelites at that time resisted the Lord.
When we realize that the Greeks are seeking Jesus during the Feast of Passover & Unleavened Bread, we then come full circle that the Lord "was found by those who did not seek me; I became manifest to those who did not ask for me" (Rom 10:20, which cites Is 65:1). These Greeks find Jesus in the wilderness of Israel, where the occupying Romans will crucify the King of the Jews, who is the first-born son (Paschal Lamb). They find Jesus because Jesus is "lifted up" on the cross from where he will call the Greeks directly to himself.