This question has come up. Quite honestly, I think it is a good question, since I had really taken it for granted. So, I grab my cup o' Joe, and begin talking to Jesus about it. I put forth the following as my argument for our responsibility to vote:
Psalm 24.1; Psalm 2.1; Matthew 28.18-20
The earth is the Lord's. All men should submit to His ways of taking care of His earth. Yes, even the pagan man will be held responsible for either submitting to the Lord's ways, or rebelling against them. Dominion and the gospel means that every sphere of life should be placed under the authority of Jesus Christ. If I am the head of the County Transportation Committee, that is my goal. If I am the Fire Chief, that is my goal. Along the way, I share the gospel and call men to repent of their unwillingness to govern the earth according to God's Word. Proposition: All men should govern the earth for God's glory
God established the civil authorities to punish evil doers. The rod is for the house, the keys (church discipline) are for the church, and the sword is for the civil government. Proposition: Civil government is God-ordained to restrain evil.
1 Timothy 2.1-3
God has given us the responsibility to pray for our leaders, that we would enjoy certain liberties. When righteous men are put into office, people rejoice (Proverb 29.2). Prayer assumes a willingness to respond to God's will (John 15.7-8). Since we enjoy the liberty of voting for righteous men, it would seem to me to be a contradiction to pray for righteous leaders, and yet not vote for them when we get the chance. Proposition: Christians are responsible to pray for and vote for righteous candidates.
Since God created the earth, and commissioned man to exercise dominion for His glory, we are responsible to press the authority (share the gospel) of Christ into every area of life.
Since Civil Government is established by God, it is a legitimate sphere. As Christians, we are to be salt and light in that sphere, calling men to submit to Christ's authority in that realm. This alone entails a willingness to serve in that realm, if the Lord leads (ex: Jonah 3.6; Acts 8.27, etc.).
Since a Christian's desire is to see Jesus exalted in every area of life, he should pray for the righteous to lead the nation, and serve for the glory of King Jesus, if the Lord should so lead.
Since we should pray for, and serve in that capacity, we should also vote for righteous leaders.
*** Side note. It is difficult for me to conceive of teaching my children American history without teaching them the legitimacy of the revolution. I believe our nation's founders had a biblical worldview that entailed the responsibility to establish a republic like ours. I am still learning about the reasons for the revolution. This only points out how far behind THEM we really are. For them to see biblical principle sufficient to go to war... and yet I can not see biblical principle for men to vote... hmm...