What you are doing is done all the time, and there is no problem with it, if done properly. The fact is these 3 prong receptacles still exists in many older homes, and there is no requirement to upgrade an entire circuit simply to plug in a device. If you go out and purchase a new electric dryer, the seller will ask if you have a 3 or 4 prong receptacle. They will then sell you the appropriate cord, based on your answer (if you don't know, they'll probably sell you both and tell you to return the one you don't need, or offer their instillation services).
RTFM
If you check the manufacturers installation instructions, there should be a procedure for connecting the dryer to a 3 prong receptacle. The procedure can very from manufacturer to manufacturer, so make sure you check the documentation for your specific dryer.
From a random whirlpool manual:
This dryer is manufactured ready to install with a 3-wire electrical
supply connection. The neutral ground conductor is permanently
connected to the neutral conductor (white wire) within the dryer. If
the dryer is installed with a 4-wire electrical supply connection,
the neutral ground conductor must be removed from the external ground
connector (green screw), and secured under the neutral terminal
(center or white wire) of the terminal block. When the neutral ground
conductor is secured under the neutral terminal (center or white
wire) of the terminal block, the dryer cabinet is isolated from the
neutral conductor.
Click for larger view
Click for larger view
Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.
Definitions:
Grounded (neutral) from the service
A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.
Grounding Electrode Conductor
This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.
Bonding Jumper
The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.
Assumption:
Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.
So you have two options:
Terminate the grounded (neutral) from the new circuit to the grounding bus.
Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.
Additional Information and Code Compliance:
Number of Conductors
Since this is a new circuit, it has to be installed to current code standards.
National Electrical Code 2011
ARTICLE 250 — GROUNDING AND BONDING
VI. Equipment Grounding and Equipment Grounding Conductors
250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.
You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.
Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.
Size of Conductors
You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.
Best Answer
If your dryer is set up for natural gas, then it isn't going to run on propane - you'll need to install a propane conversion kit. Most manufacturers sell these for specific models, and I'm sure there are some generic after-market kits out there too. These generally aren't that difficult to install - you'll basically be changing the limit in the pressure regulator and the size of the orifice to the burner.