The answer, broadly is no, handheld GPS units can not be used for IFR RNAV routes (or approaches for that matter). To know if a specific GPS unit is legal for IFR navigation you need to check up on a variety of paperwork, thats covered in this answer. Strictly speaking if you could find a certified unit it would be legal but there are no current units (to my knowledge or searching) that are.
If we look at the regulations and bounce around a bit we can see why. AC 90-108 tells us about the use of RNAV systems and states that one bust be compliant with installation instructions in AC 10-138D of which there are many different regulations. The AC does not actually specifically state that the unit must be panel mounted but in a round about way it does:
The definition of display which is used widely in the document does state it must be mounted to the panel,
Display. Within the context of this AC, a display is a device mounted
in the aircraft instrument panel to convey course guidance navigation
information to the aircrew for controlling the aircraft along the
intended route. Displays may also be used to convey other information
There are also lots of regulations about wire mounting to avoid issues and failures as well as other mounting regs. If you are interested its worth going through the document.
See Pondlife's comment bellow for more info on panel mounted GPS'
just normal en-route navigation over fixes defined by coordinates
only. That way, the GPS is not critical—if it fails, I can always
request vectors or rerouting over a navaid.
In order for the GPS to be non critical you must fly at altitudes serviced by other nav-aids. The reason GPS routes are coming into use is that they, in some cases, allow flying at lower altitudes that may be beneath ATC radar or out of range of nav-aid coverage. In the case of a failure of a GPS unit you would then be with no means of navigation and terrain avoidance.