Should an emergency field landing be performed with landing gear down or up?
There are various factors here: 


Gear acts as a speed brake, in some emergency situations if you need the added drag as you approach the field (to make the field) a gear down situation may be called for regardless of ground condition. On the flip side if the field is on the edge of gliding distance you may want to keep the gear up to improve glide. This is a judgement call during the emergency. 
Gear will provide you limited steering on the ground which for what ever reason you may need. In a gear up situation you are sliding to a stop. 
Gear will provide you brakes, if the field is short you are going to want this option. 
If something is critically wrong with the plane and you need to do an emergency gear deploy there may be no time during the emergency.  


Like anything, it depends and it will be a situational decision with merits to both aspects. You should consult the POH for your air frame as they will most likely have check lists for both scenarios. 

For example in a Piper Arrow they offer a checklist of a gear up emergency specifically: 


  Gear up emergency landing: 
  
  In the event a gear up landing is required,
  proceed as follows: 
  
  (a) On aircraft equipped with the backup gear
  extender, lock emergency gear lever in "Override Engaged" position
  before airspeed drops to 115 mph to prevent landing gear from
  inadvertently free falling. 
  
  (b) Flaps as desired. 
  
  (c) Close throttle
  and shut off the master and ignition switches. 
  
  (d) Turn the fuel
  selector valve to OFF" 
  
  (e) Contact surface at minimum possible airspeed.


Here is what happens in a small plane when the gear collapses on a grass field. A gear up landing is not an automatic hull loss, in many cases they can be repaired. Considering the age and use level of most of the countries GA fleet you may be surprised to find out just how many planes have actually been through a gear up landing. 

I was always instructed during my training that ditching gear down (or with a fixed gear) in water was not really viable. There seems to be some info indicating that there is little data to support this. Of course you will find endless debate on the internet about it and some answers that fail to provide sources. Some of the misconceptions may be fueled by what happens when a float-plane digs in on landing.