Being I am not a Calvinist, I am unfamiliar with the acronym T.U.L.I.P. So my reply is based off of what I got from a quick search.
Total Depravity: This assumes that spiritual death is a cessation of the spirit. If this was true, then I would agree with total depravity. However, I do not believe spiritual death to be the cessation of the spirit, but rather the spirit going into a dormant state where it is no longer sensitive to God's presence. But, it is still capable of responding to the convicting work of the Holy Spirit. If our spirit was actually dead, then even the convicting work of the Holy Spirit would be of no use, and we would remain lost.
Also, and I can't say this with absolute certainty, because we are made in the image of God, I believe the cessation of any part of our being: body, soul, or spirit would result in what looks like physical death. God is revealed to us in three persons, but all are still one God. Being in the image of God we too are made up of three parts, as I just described. These three parts are what makes us human. This would explain why in the new age, we receive new physical bodies. Not only to maintain the image of God, but because we cannot exist without all three parts.
Unconditional Election: I heard TD Jakes say once that God choosing to save whomever it is He chooses to save is a great display of His grace because no one deserves salvation. Rather than lose everyone, God chooses to save some. To me this doesn't paint a very attractive picture of God. If God is capable of saving some, then He is clearly capable of saving all, yet chooses not to? This does not sound like the God presented to us in the bible.
Limited Atonement, Irresistible Grace, and Perseverance of the Saints: These next three are all based of the premise that it is our sins that separate us from God. That salvation is found in getting our sins forgiven. This same premise applies to almost all forms of Christianity being preached today.
Ours sins are not the problem. They are not what separates us from God. Our sin is a form of behavior. Our sinful behavior is a symptom of our condition, not the cause. Our condition is that of spiritual death. Spiritual death was the punishment Adam and Eve received for their disobedience. Because spiritual death plus spiritual death will always equal spiritual death, we've all come into the world already spiritually dead (separated from God). The symptom of our condition, being separated from God, is that we sin.
Now because it is not our sins that separate us from God, getting our sins forgiven isn't going to save us. We are saved when we place our faith in God for salvation. The forgiveness of sins is what makes salvation possible. Everyone from the cross forward has come into the world already forgiven of their sins (this was made know when God tore the temple curtain). Salvation however (which is found in the resurrection), only occurs once we place our faith in God. So now if God has forgiven all of mankind, then clearly salvation must also be available to all of mankind.
When the Holy Spirit convicts us, it is not of our sin, but of our unbelief. All sin is forgiven, but our unbelief isn't. This we must choose to repent of.
Because the punishment for all sin was forgiven at the cross, there remains no more punishment to be handed out by God. Because there is no more punishment left to be handed out, there is no longer any possibility of spiritual death, separation from God. Because spiritual death is no longer a possibility, there is no way to lose your salvation, or to wilfully lose it by walking away since you can't wilfully choose to spiritually die. That is in God's hands alone.
Having said all this, in the end, I honestly don't think it matters what one believes in terms of salvation, but rather who we believe in for salvation. Anyone who places their faith in God for salvation will be saved, regardless of whatever else they choose to believe.
Now before anyone comes at me with their nickers in a knot claiming that John 14:6 says we must believe in Jesus to be saved... let me be the first to remind you that Jesus is God! Not simply a means too God. He established the fact that He is God in the first part of the verse. Therefore, He is not saying no comes to the Father/God/Me except through the Father/God/Me. This makes absolutely no sense. Why do you need to go through God in order to get too God when going through God means you're already with God?
What Jesus is actually saying is that regardless of how you choose to come to God, you will have come through Him. He is the one that made salvation possible. It is only through His forgiveness of sins that anyone can now come into the presence of God for salvation.