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.
This is an unfortunately-all-too-common wiring error, even without a 12/2/2 cable to further confuse things. Most folks (even electricians) think "I'll just nut the neutrals together" since that's what works when everything in a box comes from one branch circuit, which is the case 99.9% of the time -- in the 0.1% case, such as yours, though -- this rule leads to a looped neutral, which violates 300.3(B):
(B) Conductors of the Same Circuit. All conductors of the same circuit
and, where used, the grounded conductor and all equipment grounding conductors
and bonding conductors shall be contained within the same raceway, auxiliary
gutter, cable tray, cablebus assembly, trench, cable, or cord, unless
otherwise permitted in accordance with 300.3(B)(1) through (B)(4).
and 310.10(H)(1):
(1) General. Aluminum, copper-clad aluminum, or copper conductors, for
each phase, polarity, neutral, or grounded circuit shall be permitted to be
connected in parallel (electrically joined at both ends) only in sizes 1/0 AWG
and larger where installed in accordance with 310.10(H)(2) through (H)(6).
This is easy to fix, though -- make sure that the neutrals in your 12/2/2 cable (white and white/red striped, or "candy cane wire" ;), are not nutted together anywhere.
Best Answer
Normally neutral and ground are bonded at your main panel, but must not be bonded at any subpanels downstream. So mixing neutrals and grounds on the same bar is OK in the main panel but not at subpanels.
The 14/3 with black and red on two breakers and white on the neutral bar - that's a multiwire branch circuit with a shared neutral. When installed properly this is code compliant. However take particular care when working on and around this circuit, if the neutral is disconnected it can result in an overvoltage on the affected circuits.