In many ways, the play seems irrelevant to a modern secular audience. We don't talk about "sin" and we don't spend much time worrying about the afterlife. More people don't go to church than do, and of those that do, the hell and brimfire notion of a God who demands that we fear Him is alien. So there is definitely (thank the gods) something of a cultural divide between our lives and those of the average medieval peasant.
Having said that, we are mortal. We are a culture obsessed with worldly pleasures, and, until we come face to face with loss, we forget, like Everyman, that we can take precious little with us when we go. So what do we leave behind us? Our friends, our family, our material wealth and, if we are lucky, a few people who remember something good we have done.
From a Christian perspective, the notion that good deeds can speak for us at the gates of heaven is outdated--most denominations of Christians would say that "grace" rather than "deeds" will open the doors. From a humanist perspective, however, deeds are the key to a well-lived life, and really that was what morality plays were all about--they were 'in your face' reminders that there is more to life than sensory pleasure, friends and family, and that what gave life purpose and meaning was more what we've done for others than what we've cultured or accumulated along the way.
Everyman would apply to our time in a general way, and perhaps in a more specific way. In a general way, all humanity must come to terms with being mortal, and with the challenges of living moral lives. We will all die, and while we're alive, we must all act morally. This play offers one answer to those questions, and dramatizes it in a very clear fashion. We can look at this historically, and ask, "Ah, that's how they answered these questions; how do I /how do we answer them?"
The more specific answer would be if one were a Christian. In Christianity, the nature of the world essentially doesn’t change. Therefore, the play would be immediately, directly, and specifically relevant: you have to make the right choices or go to hell. In fact, given the emphasis on wealth today, Goods would be a huge character, and its betrayal would hurt even more.