Can anyone tell me an efficient, organic way of stopping bermuda grass from taking over my garden?

Sure, get a potato fork and start digging out the roots.

You can also solarize the area by putting a big sheet of clear plastic down on the garden area. Get the soil wet, spread out the plastic and dig in the edges so you get a tight fit (should be able to bounce a coin off of the plastic). Let this sit through the hottest time of the year (June/July in the Northern hemisphere) for about 8 weeks and most of the grass should be killed.

After that you will need to put in a subterrainian barrier that goes down several feet to keep the roots from reinvading and you will also have to dig out roots regularly as the grass seeds germinate.

Any garden plot that has been in proper organic production for 5+ years will have less and less of a grass problem as the soils are transformed to be more friendly towards crops and less friendly towards grasses. Often having a grass problem is a sign that you have either fertility problems or compaction problems so you may want to get a soil test done to see for sure.

Remember, if you use herbicides you have to constantly reapply them to keep the weeds at bay. With organics you do the same thing but with elbow grease instead of toxic chemicals.

Elbow grease and a sharp hoe. The roots have to be kept at bay. Remove any and all that you can and do not let any more grow. Continue to remove it and eventually, believe it or not, you will get rid of it. Turning the soil over before a hard freeze will help too, exposing the roots to freezing temperatures.

Mulch, mulch, mulch...and then if runners come up through the mulch, pull them out by the roots. Eventually the b/grass will go away. At least it will become less and less...Sundown