In (at least) pop-Calvinism, there is a tendency to view man as essentially sinful, that is, down to the very structural core of his being, man is inherently and inescapably controlled by sin. Man, in other words, cannot do anything to please God, whether towards earning salvation (understood as monergistic justification) nor towards maintaining or advancing the relationship (a sort of monergistic sanctification). While I haven't found this understanding to necessarily be the orthodox and official interpretation of the Calvinist system, it holds a lot of traction at the "lay" level and amongst certain types of preachers and pastors. I cannot tell, statistically speaking, whether or not this is a majority opinion, but I hear it often enough from a variety of sources that I think it should be taken seriously.
However, I think this understanding (which, at least superficially, sounds similar to Luther's in Bondage of the Will and Calvin in On the Secret Providence of God) is fatally flawed because it misunderstands the necessity of distinguishing between two distinct definitions of the word "nature." The first definition of nature I'd like to term "essential nature." This understanding of nature is that it is what man is created to be on the structural level by God: it is what makes mankind mankind, as opposed to any sort of created reality. It is what all humans share that makes them describable as such. The common theological understanding of this is the imago Dei, or Image of God. We all share (or participate) in God's Image (which the New Testament reveals to us as Jesus Christ) to some degree or another. What this means, though, is hard to define precisely: does it mean our rationality? our responsibility and tendency towards rule? our community one with another? All of these are theories that have been held throughout Church history (with St. Augustine, Richard Middleton, and Karl Barth being respective representatives). Whether or not our essential nature can be reduced to any one theory is debatable: our Image-bearing reflects the Image Himself, who is the Icon of the Invisible and Infinite God. Our bearing, then, is capable of the same infinity (here is the basis of the Patristic doctrine of theosis). This nature is fundamentally, by decree of God, "good" as proclaimed in Genesis 1. There is no language, that I can find at least, that describes this Image-bearing, or essential nature, as distorted, cracked, destroyed, mangled, or any other such perjorative term: even with the ravages of corruption and sin, we remain essentially human, bearers to a greater or lesser degreee, of God's Image. God's statement to Noah that man is still in the Image of God, immediately after the judgment of the Deluge, goes a long way towards showing this. This is also the basis of our redemption as well, since this Image God does not desire to see destroyed. (Questions about how our structural integrity before God, regardless of our directional infidelity, and the "eternal decree" of double predestination will have to wait until a different time; questions there are, but my ability to adequately answer them is another question altogether).
In the essential sense, then, it is incorrect, and quite possibly blasphemous, to say that mankind has a "sinful nature." God did not create man's nature to be sinful, nor can we so attribute power to sin and death that they are able to thwart the primal command of God's Word to structure human reality. Sin and death are more powerful than man outside of God's grace, but not ever more powerful than the Creator and Redeemer of the cosmos.
The second definition, however, allows for an understanding of man having a "sinful nature" in a proper sense. This is term "energetic nature," that is, a description of the actions (or energies) of man, which include the tendency of the will towards evil and sin and death. In other words, man often acts and thinks and speaks in sinful ways. A man can be an Image-bearer and yet be the worst cur on the face of all Creation. The effects of Adam's corruption on mankind have made such "modes of action" seem normal or, to use the term at issue, natural. This sort of corruption of our energies can seem to effect what we consider structural elements of mankind (our biological makeup, our sexuality, our personalities), however these things are distorted activities that can be healed by the presence of the Holy Spirit in conjunction with our own ascesis. We are in the environment of death, of being separated through Adam's rebellion from the Life of God, so our actions tend towards that way, even if our essential nature is proclamed "good" by God Himself. This is also what makes sin so sinful, as St. Paul might say: mankind, which is supposed to be oriented towards God and partaking of His Life (the Holy Spirit), has instead turned towards the nothingness of sin and death. This is why Christ came to "condemn sin in the flesh" not to condemn humanity. Man has become, through his federal head of Adam, a slave to sin, death, and the devil; he acts as a servant of such, even though his original intent is towards the freedom of Life in and with God. These energies need to be redeemed, which is why it is necessary for Christ, in the Incarnation, to take on a full human nature (essential good, yet subject to corruption), including these energies, so that we might wholly and fully be redeemed in Him.
To use a Biblical expression, we might term this the "likeness of God." The Image we bear whether we will or no, the likeness, however, is not mentioned after the Fall. Our likeness to God, promised at the Creation, was thwarted by Adam's desire to be "like God" (Gen. 3:5) on his (or the serpent's) own terms, rather than God's. Christ restores us to a place where we can, again, be like God -- as He says in the Sermon on the Mount ("Be perfect as your Father in heaven is perfect").
What, then, of total depravity? Our energies, our actions, our tendencies are, by virtue of their participation in the corruption of Adam, oriented away from God, rendering us inable to participate in God's Life. In this sense, yes, we are totally depraved: our environment and our energetic natures are estranged from God, even when we seek -- outside of God's grace -- to do good, we do it corruptly: one only has to look at the corrupting effects of nationally-instituted charity to see this. However, in Christ, although we still often fail, we can do good, that is, we can share the Life of God, through our actions in the Spirit. This does not mean, however, that we "earn" God's favor or love, the Scriptures never use this sort of language. Instead, we do what we were created to do as God's Image-bearers, as icons of Christ, when we do good. We are most fully human when we participate in God's Life and share that Life with the rest of the Creation, both human and non-human.