Protestants do not see that Christ ever instructed his followers to pray (only to baptize) "in the name of the Father, and of the Son, and of the Holy Spirit". On the other hand, Christians are repeatedly called to invoke the name of the Lord:
To the church of God in Corinth, to those sanctified in Christ Jesus and called to be his holy people, together with all those everywhere who call on the name of our Lord Jesus Christ ...
(1 Corinthians 1:2a, New International Version)
You did not choose me, but I chose you and appointed you ... so that whatever you ask in my name the Father will give you.
(John 15:16, New International Version)
"Let everyone who calls upon the name of the Lord avoid evil."
(2 Timothy 2:19, New International Version)
Thus, there is solid Scriptural evidence that the early Christians directly invoked, and were instructed (directly or indirectly) to directly invoke, the name of the Lord, rather than speaking the names of the Trinity; the only occurrence of the phrase "the name of the Father" refers to the "Great Commission" passage in Matthew:
"Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, ..."
(Matthew 28:19, New International Version)
Jesus, furthermore, is the only one to whom humans need turn in order to approach God. This is pounded into the heads of Christians especially in the Letter to the Hebrews:
... Jesus the mediator of a new covenant ... (Hebrews 12:24)
For this reason Christ is the mediator of a new covenant, that those who are called may receive the promised eternal inheritance—now that he has died as a ransom to set them free from the sins committed under the first covenant. (Hebrews 9:15)
And most famously
For there is one God and one mediator between God and mankind, the man Christ Jesus, who gave himself as a ransom for all people. (1 Timothy 2:5–6a)
He is not only mediator, but mediator in the sense that the Jewish high priests were mediators between God and the nation of Israel:
We have this hope as an anchor for the soul, firm and secure. It enters the inner sanctuary behind the curtain, where our forerunner, Jesus, has entered on our behalf. He has become a high priest forever, in the order of Melchizedek. (Hebrews 6:19–20)
(All quotations from the New International Version)
Thus many Protestants, following the example of Scripture, feel that it is far more appropriate to invoke the assistance of God by calling upon the name of Jesus.
I should note that not all Protestants follow this example all the time, or perhaps even much of the time. For example, the website of the United States Conference of Seventh Day Baptists contains a prayer for the sick and shut-in which ends "These things we pray in the name of the Father, the Son and the Holy Spirit." Thus, perhaps the question should be phrased "Why do some Protestants say ..."
Catholics, on the other hand, use "in the name of the Father ..." as part of "the Sign of the Cross"—a sacramental (that is, a sacred activity which resembles a sacrament; see Catechism of the Catholic Church paragraphs 1667–1668). In this sacramental, the Catholic makes the outline of a cross on his body: he touches his forehead as he says "In the name of the Father...", then his chest or stomach, saying "... and of the Son, ...", then each shoulder in turn, saying "... and of the Holy Spirit. Amen."
The Catechism explains:
The Christian begins his day, his prayers, and his activities with the Sign of the Cross: "in the name of the Father and of the Son and of the Holy Spirit. Amen." The baptized person dedicates the day to the glory of God and calls on the Savior’s grace which lets him act in the Spirit as a child of the Father.
This is not done as a Biblical command, but as a reminder of the One who saved us by his death.