I thought the NT taught that everyone, no matter how good they tried to be, was a sinner headed for Hell - unless they accepted Jesus. Does it say otherwise?
It doesn't say either explicitly. There are many Scriptural passages regarding salvation that have to be synthesized. I posted the position of the Catholic Church above, which is a synthesis of Scripture and Tradition.
The Church distinguishes between explicit and implicit faith. A non-Catholic who serves Truth (God) to the best of his ability, without accepting Christ explicitly, through no fault of his own (ignorance or poor teaching regarding Jesus), may be saved (but his salvation would be through Christ). This would represent implicit faith.
The position you're referring to sounds Fundamentalist, but I'll let them speak for themselves.
Here are some important, relevant Scriptures:
John 15There are many others, but these are some of the more difficult passages.5"I am the vine; you are the branches. If a man remains in me and I in him, he will bear much fruit; apart from me you can do nothing. 6If anyone does not remain in me, he is like a branch that is thrown away and withers; such branches are picked up, thrown into the fire and burned. 7If you remain in me and my words remain in you, ask whatever you wish, and it will be given you. 8This is to my Father's glory, that you bear much fruit, showing yourselves to be my disciples. 9"As the Father has loved me, so have I loved you. Now remain in my love. 10If you obey my commands, you will remain in my love, just as I have obeyed my Father's commands and remain in his love. 11I have told you this so that my joy may be in you and that your joy may be complete. 12My command is this: Love each other as I have loved you. 13Greater love has no one than this, that he lay down his life for his friends. 14You are my friends if you do what I command.
James 2
14What good is it, my brothers, if a man claims to have faith but has no deeds? Can such faith save him? 15Suppose a brother or sister is without clothes and daily food. 16If one of you says to him, "Go, I wish you well; keep warm and well fed," but does nothing about his physical needs, what good is it? 17In the same way, faith by itself, if it is not accompanied by action, is dead. 18But someone will say, "You have faith; I have deeds." Show me your faith without deeds, and I will show you my faith by what I do. 19You believe that there is one God. Good! Even the demons believe that—and shudder.
John 3
5Jesus answered, "I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit.
Matthew 7:21
"Not everyone who says to me, 'Lord, Lord,' will enter the kingdom of heaven, but only he who does the will of my Father who is in heaven.
John 6
53Jesus said to them, "I tell you the truth, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. 54Whoever eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day.
Acts 17:23
For as I walked around and looked carefully at your objects of worship, I even found an altar with this inscription: TO AN UNKNOWN GOD. Now what you worship as something unknown I am going to proclaim to you.