According to Orthodox Christianity, did God die on the cross at Calvary?

The concept Communicatio idiomatum was developed before Great Schism (1054) and therefore may be considered as teaching of The One True Holy Church of Christ (currently referred to as Orthodox).

(1) Human properties can be attributed to Christ as God:

"crucified the Lord of glory" (1 Corinthians 2:8)

"Therefore take heed to yourselves and to all the flock, among which the Holy Spirit has made you overseers, to shepherd the church of God which He purchased with His own blood" (Acts 20:28)

"The first man was of the earth, made of dust; the second Man is the Lord from heaven" (1 Corinthians 15:47)

"We were reconciled to God through the death of His Son" (Romans 5:10)

(2) Divine properties can be attributed to Christ as man:

"No one has ascended to heaven but He who came down from heaven, that is, the Son of Man who is in heaven" (John 3:13)

Thus according to Communicatio idiomatum Transcendent and Eternal God die on the Cross as man.

From the Orthodox POV, the answer is unequivocally yes.

The Orthodox understand John 1:14 literally:

And the Word became flesh and dwelt among us, and we beheld His glory, the glory as of the only begotten of the Father, full of grace and truth. (NKJV) [Emphasis mine]

The Word of God, the Second Person of the Trinity, became flesh. Not had flesh. Not wore flesh. Became flesh. The Orthodox Study Bible attaches the following note to John 1:14

The Word became fully human without ceasing to be fully God. He assumed complete human nature: body, soul, will, emotion, and even mortality -- everything that pertains to humanity except sin. [Emphasis mine]

The Word assumed even our mortality. This is expressed in the Symbol of Faith (Nicene Creed)

[We believe] in one Lord Jesus Christ, the Son of God, the Only-begotten, begotten of the Father before all ages; Light of Light: true God of true God; begotten, not made; of one essence with the Father, by Whom all things were made; Who for us men, and for our salvation, came down from the heavens, and was incarnate of the Holy Spirit and the Virgin Mary, and became man; And was crucified for us under Pontius Pilate, and suffered, and was buried; And arose again on the third day according to the Scriptures;

Jesus Christ, true God from true God, ... became man ... was crucified for us under Pontius Pilate, and suffered, and was buried. It is hard to understand that in any way other than 'God died.' There is no change in subject, no way to separate a god-soul from a man-body. The Only Begotten Son became man and died.

From the Council of Ephesus (3rd Ecumenical Council)

If anyone shall not confess that the Word of God the Father is united hypostatically to flesh, and that with that flesh of his own, he is one only Christ both God and man at the same time: let him be anathema.

The flesh is his, the Word of God, own. You asked

I accept fully that Jesus the man indeed died on the cross, but did Jesus the God die also?

Orthodoxy does not permit this division. There is no Jesus the man and Jesus the God. There is only Jesus Theanthropos, Jesus the God-man. The Council of Ephesus forbade ascribing some attributes to Christ's humanity and other to His Divinity. That is, the mortality that Christ assumed from humanity must be applied, through the Hypostatic Union, to his Divinity as well. Again from Ephesus:

If anyone shall divide between two persons or subsistences those expressions (φωνάς) which are contained in the Evangelical and Apostolical writings, or which have been said concerning Christ by the Saints, or by himself, and shall apply some to him as to a man separate from the Word of God, and shall apply others to the only Word of God the Father, on the ground that they are fit to be applied to God: let him be anathema.

As part of the Liturgy of St John Chrysostom, (the weekly worship service), the priest prays in reference to the Bread and Wine, Body and Blood:

Tremble, O man, as you behold the divine Blood. It is a burning coal that sears the unworthy. The Body of God both deifies and nourishes me: It deifies the Spirit and wondrously nourishes the mind.

The Body and the Blood are clearly considered divine.

Consider also anathemas of the Second Council of Chalcedon, which the Orthodox accept as Ecumenical and binding:

Anathema III: If anyone says that God the Word who performed miracles is one and Christ who suffered is another, or says that God the Word was together with Christ who came from woman, or that the Word was in him as one person is in another, but is not one and the same, our Lord Jesus Christ, the Word of God, incarnate and become human, and that the wonders and the suffering which he voluntarily endured in flesh were not of the same person, let him be anathema.

That is, God the Word suffered in the flesh. From the Anathema IV it is clear that the Orthodox consider the flesh of Christ to be united to the Divinity of God the Word:

But the holy Church of God, rejecting equally the impiety of both heresies, recognizes the union of God the Word with the flesh according to synthesis, that is according to hypostasis. For in the mystery of Christ the union according to synthesis preserves the two natures which have combined without confusion and without separation. [Emphasis mine]

The Orthodox can say without question that God died. Note, however, that this in no way implies that God ceased to exist or ceased to function. However, He, through the Hypostatic Union, suffered death, having His Soul separated from His Flesh.