>> near the bottom
The word "near" is key. If there is any spot lower than your bed, that is close enough that you can dig a slit trench from the lowest part of your new bed to the lowER spot that you've located, water will drip from your bed into your trench, then run down to the lower spot and leave your bed well aerated.
If the lower spot is in a neighbor's yard, console yourself that you're only helping it run off faster. It would always have gotten downhill eventually. However, you may be shifting your temporary sogginess into his temporary sogginess, depending on how well graded and drained his yard is.
The key question for both of you is where the water goes after it reaches the lowest local spot? If that spot has good drainage, all is well.
Another way to think of this is "lowering the water table quickly after a rain". If your trench exports water, the water level in soil surrounding and uphill from your trench will rapidly drop to be level with the floor of your trench. How far the rapid influence of your drainage extends depeneds on soil density. In very heavy clay, it might extend only a few feet to either side of your trench.
Even a narrow slit trench is plenty to drain anything short of a monsoon. Mowing may be easier and ankle-breaking harder if you backfill the slit trench with drainage gravel (French Drain). Or, infinitely cheaper and lighter, make the trench wide enough to hold some plastic drainage pipe (perforated, corrugated pipe, 4-6" in diameter, or you may find something smaller).
Of course, the key point to a drainage trench is that the floor of the trench needs to slope DOWN consistently. The easy way to make that exact is to dig it out roughly, then wait for a heavy rain. Puddles will show you the low spots. You can use a hoe or mattock to "spread the floor around", or a shovel to dig deeper where the trench was not deep enough.
There are also things you can do with string and stakes or a landscape level, or even a laser pointer to get a uniform grade, but I just pick a path and dig, then wait for rain.
