The reduction of the third commandment to an instruction about using coarse language is one of the telltale signs of legalism throughout the history of the Christian church and beyond. Ever since Cain, mankind has sought to maintain a superficial righteousness when faith and love were not present.
The Jews, whom the Old Testament describes as having hearts that were far from God throughout most of their history, went to great lengths not to utter or write the Tetragrammaton (YHWH). But after He announces His "name" to Moses (and the world), in Exodus 3:14, He goes on to say in verse 15:
God, furthermore, said to Moses,
“Thus you shall say to the sons of Israel,
‘The Lord, the God of your fathers,
the God of Abraham,
the God of Isaac, and
the God of Jacob,
has sent me to you.’
This is My name forever, and
this is My memorial-name to all generations."
Thus, bracketed between "furthermore" and "This is my name forever," is a reference to His behavior in history toward His people with whom He had a relationship. This can only be understood when we understand the greater meaning the Hebrew culture put on the words "name" and "called." God named things as He created them, forever proclaiming truths about them. After naming the animals, Adam named Eve based on an observation he made about her. Starting with Eve and continuing throughout much of the Old Testament, we read about people, naming their children based on something that was descriptive of relevant events in their lives (and occasionally God would name them based on what was going to happen). In Isaiah 7:14, and again in Matthew 1: 23, we read a predictions that Jesus will be called "God is with us" (or "Immanuel"). In Isaiah 9:6 we get more names: Wonderful, Counselor, Mighty God, Eternal Father, Prince of Peace. This continues in the New Testament with the renaming of Simon to Peter, and with the many mysteries that are "called" by their "names" in the Book of Revelation.
The superficially-minded, the legalistic, dead religions, and cults see the names of God as rules of acceptable words, and words of mystery and magic. The faithful and righteous see these as descriptions of the eternal nature and loving character of the one true God. They see it this way because God's love has prevailed upon them, because they have put their trust in that love, that love which He expressed to all on the cross. They see it this way because these "names" are coming out of their own mouths, from their hearts, describing God's character as exhibited in His Son.
So what does all of this have to do with the third commandment? God has invested a lot in communicating His nature and character to mankind, creating for Himself a people through His actions in history, that He might display His character, and they might record it and preserve it in scripture for the world. Jesus completed this display, proving God's character to the world for all time through His suffering and death on the cross, entrusting the message to His apostles for the whole world for all time.
To "take God's name" means to call yourself something with relation to God, such as "God's people," or "God's children." The word "vain" means "empty," "without productivity," or "without bearing fruit."
If the Jews took God's name upon themselves, calling themselves "God's people," but did not keep His commandments, they were, according to the usage of the language at that time, "taking God's name in vain."
Likewise, for someone to call themselves a "Christian" without being transformed by God's love to be obedient from the heart, is to take God's name, "Jesus Christ," upon one's self in vain.
And this also helps us understand the same type of usage of the word "name" in another context—that is, when we speak of the authority of a name. A priest is one who intercedes between man and God to bring man into, or keep man in, a right relationship with God. Jesus is our high priest forever (Hebrews chapters 4-11), because His expression of God's love toward us has once and for all performed the necessary action to melt our hearts, and to bring us to, and keep us in, a loving and obedient relationship with God. There is not, and never will be, a need for any other kind of priest. He's done it all, and there is nothing more that anyone can do (thus, "there is no other name under heaven that has been given among men by which we must be saved").
So when we are submitted to God, through Jesus (remember, with meaning, not with magic words), we have His authority, each and every one of His children, just as an agent of the government (a police officer, for instance) has the full authority of the government, as long as he is in good standing and acting within the scope of his duties.
And again, when praying in Jesus' name, we pray while living according to His will, and we speak according to His will, and it will be done. We don't say magic words at the end of our prayer and think it will have some effect; this is like witchcraft, not a relationship with the one true God who has revealed His name (read "character") throughout history.
"Little children, let us not love with word or with tongue, but in deed and truth." (1 John 3:18)