There are two ideas here: 1) Salvation 2) Grace.
Evangelicals have made this word refer to "being saved from hell". In biblical usage "saved" had a more general, this-worldly meaning. Israel was saved from captivity, for example. When the NT talks about salvation, it is a really big assumption to say it's talking about salvation from hell (especially seeing as I doubt they even had the modern concept of hell, which I think was a Persian/Greek idea).
It seems clear that in the context of many verses, the NT writers mean that God has saved us not from hell, but from the harm of sin: sickness, oppression, immorality, and even death (think of some people Jesus healed, for example).
Furthermore, the early Christians kept the Jewish tradition that final judgment was in fact based on our character and behavior (I'm not going to prove this here). In Jesus, they saw that God had shown them how to be and live in such a way that was favourable to God. In other words, they believed that Jesus had shown them that if they follow his way of life, they would receive a favourable final judgment.
And here is where grace comes in. You see, they had a problem. Not only did they not know how to live in a way that pleased God, but they needed help to change. In Jesus, they saw God as graciously coming and helping them while they were still sinners - to help them become righteous, to free them from sin and show them give them a new way of life. He didn't have to help them. They didn't deserve his help, but he helped them anyway. Because of all this and their new way of life, they could be sure that he had indeed forgiven their sins. God not only forgave them, but he freed them from sin. God was graceful in an active, helpful way... not simply a passive "I've not send you to hell" way.
This concept of grace becomes even more powerful when we realize that it is exactly this kind of grace that Jesus taught us to display to others. Just as Jesus showed grace to free sinners from sin, we are to show grace to free more sinners. We are to carry on his work, his mission and ministry to further his vision - the Kingdom of God. So it is not merely God who shows grace, but it should be followers of Jesus also. The kind of grace is exactly the same.
In these ways, I can affirm the biblical concept of judgment based on our character and behaviour (not mental belief that Jesus took away our sin on the cross) and I can also affirm the passages that speak of being saved by grace. We are saved the grace of both God and his human agents in the world from sin (ours and others') into a righteous way of life, and this is here, now, in the present. As a result, we become righteous. God wants heaven to be for righteous people - and that’s what we become because God has graciously led us to that way of life. We can now stand in God's GRACE (i.e. favour) because he has helped us to be people who do actually please him (the NT authors used grace both in the sense of "a favour" and also in the sense of being "in someone's favour").
Let me mention that we don't "earn salvation" by being righteous. Rather, God saves from our unrighteousness so that we may be righteous. Nor do we "earn access into heaven" by being righteous, because God helped us to be righteous in the first place. And being righteous doesn't even make us deserve to go to heaven, or force God to let us in - it still depends on his grace.
Let me give an analogy (which, like all analogies, shouldn't be taken too far):
Imagine only surfers will get into heaven. God's decided that heaven is just for surfers. You don't know how to surf. Tragic, you're doomed. But God is graceful toward you - he comes and shows you how to surf. Then, having then been taught how to surf and saved from not being able to surf, you can get into heaven. But you also can also surf before you die, and teach others how to have fun by surfing. And what if this was the whole point? That surfing is the ultimate thing to spend you time doing, that it was what God made us for.
Here's the parallel. Surfing is analogous to living in a righteous way, the way that is ultimately best for us, and for this reason it is the same way of living that God is pleased with.
So we are still "saved from hell" by grace, if you want to say that... it just works in a different way to the evangelical gospel. In my opinion, this is how the NT writers understood things and is a far more consistent view. It is also consistent with modern scholarship and historio-social investigations of recent years.