Q:

Risks and benefits of natural birth

Women have been giving birth naturally throughout history, but what are the risks and benefits to natural birth, i.e. avoiding epidurals and analgesics?  Is having a natural birth statistically more or less risky than getting an epidural? 

A:

Of course it can be done and it happens thousands of times every day.  
The key questions for you is probably "Is the the right choice for me and our baby?". 
This depends on a variety of factors, a lot of which are very personal, so only you can decide.  A few points

There is a fair amount if research out there on which one is "better".  Unfortunately a lot of it is very biased so it's difficult to get an objective view.  The World Health Orginazation has published http://www.who.int/maternal_child_adolescent/documents/who_frh_msm_9624/en/ . 
There are risk factors associated with home births and there are risk factors associated with clinical birth
Requirement for a home birth is that everything is "normal" and that there are no specific pre-identified risks. 
A well trained birth attendant (mid wife) is vitally important.  It's great if you can establish a good relationship with the mid wife during the preganancy. 
There ought to be an emergency plan so if things go wrong, you can react quickly. 
There is enormous cultural bias.  While a doctor in Germany yelled at us "do you want to kill your child", 100 miles across the border in the Netherlands it's a complete non-issue and home birth is very frequent. 

This being said, we had all three kids at home without pain treatment and it was a wonderful experience.  You are in control of most of the process and you are in a comfortable environment with people that you know and trust. 
That's the right thing for some people, but not for others, so it's a very personal choice. 

A:

Childbirth hurts.  It always has.  But the amount of pain you experience depends on many factors, such as the position of the baby, whether you have or haven't ruptured the amniotic membrane, your pain threshold, your anxiety level, your position in labor, and a host of other factors, many of which you have no control over.  Can a first time mother do it without pain meds?  Absolutely. 
Most physicians (and here I can only speak to the US) don't have enough training in natural deliveries to really help the mother that way (there is a reliance on drugs for pain relief), so I agree wholeheartedly with @HelenM and @Hilmar (That answer was not posted when I started to answer this, or I would simply have commented on my complete agreement with @Hilmar) to get yourself a good, experienced midwife or birth coach who could help you through it and find a breathing and relaxation method you and your partner agree on and practice ahead of time, if you have decided on that route.  Even if you have not, I would still recommend you look into it. 
This topic draws strong opinions not necessarily based in fact.  There are a great many people who are absolutely convinced that you should do it without pain medications, just because they have, or their bias towards all things natural. 
The best advice I can give you is to plan for the best and prepare for the worst.  Definitely practice breathing and relaxation, because it will help you regardless of whether you choose pain meds or not.  Have a back-up plan in case things aren't progressing the way you and your midwife would like.  The baby has to come out, and it is very highly likely that you will not be scarred for life by the experience.  If it wasn't worth the final result, people in a position to choose would only have one child. 

A:

I did it and would do it again!  I felt there were more risks with epidural and other pain relief methods than doing it the old-fashioned way.  That said, I would recommend to anyone doing it naturally to have a midwife, a doula, or a birth coach who has assisted a birth before (not your partner!),  who can advise the best positions for pain relief and know how to aid and speed labour naturally.  I also read up on relaxation exercises during labour (Bradley method, but there are others) ahead of time with my partner.