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:

1. Terminate the grounded (neutral) from the new circuit to the grounding bus.

2. 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.

This is unfortunately not as simple as it looks

You'd think you'd be able to get by with a simple direct buried 14/4 cable, and you're almost right. However, 14/4 UF is a non-starter (nobody makes it). So, you'll have to lay some Schedule 80 PVC conduit in your trench instead and pull 14 AWG THWNs (3 hots, 1 neutral, 1 ground, just as you were thinking) through that. The trench should be about 24" deep to make sure you meet the 18" minimum bury depth requirements, and should have a warning tape atop the conduit so that the next bloke who digs there gets a reminder before they get bit.

You'll want to use fat conduit (say 3/4" or 1" with reducing fittings) to make the pull not totally unmanageable, and your box inside will need to have a conduit knockout to fit the conduit (many plastic boxes don't, so you'll have to be careful when picking the box out). It's probably best to run the conduit up the outside of the wall and then have a weatherproof nonmetallic LB body going straight into the back of the box, for that matter, instead of having conduit coming into the foundation and then up through the top plate.

At the other end, you'll run the conduit up to a weatherproof nonmetallic type X body and then run a nipple from the type X body to a nonmetallic FS box atop that for your receptacle, all mounted to some scrap 2x6 that is used as a bracket attached to a 4x4 post in the ground. The two side ports run from the X body to weatherproof nonmetallic L bodies (an LL on the left and a LR on the right) which then run to conduits going back down into the ground to the light fixtures -- these conduits have one hot, a neutral, and a ground in them, with the splices being made in the X body.

Also, use different colors (anything besides white, grey, or green) for each of the hots here so that you can keep them straight in the outdoor wiring -- one of them will be an unswitched hot for the receptacle, while the other two are switched hots for each set of lights. Last but not least, make sure there's a "while in use" hood on the receptacle box -- an "extra duty" type should be used, even, to provide the best ruggedness available.