Here is nice paper on this matter: http://www.ncbi.nlm.nih.gov/pubmed/14529971
In my experience, plates for plating should have a fairly dry surface to avoid spots from mixing. Usually, I put my stored plates at 30-37C for about 30min - 1 hour prior the plating or at the bench opened (with gas fired up in the middle to prevent contamination).
Additionally, I always wait till spots get dry after plating, before covering and flipping the plates.
One more thing, freshly made plates should be dried a bit too even though they look ok - media on freshly made plates usually contains more moisture than on stored ones.
Olga is right. The plates need to be fairly dry, but not too dry (overdrying will cause ridges to form, the droplets will run along the ridges). We generally found that biohood with blower on, plate cover off, is more efficient in drying the plates, normally 30-45 minutes for "older" plates. Different media will need different drying time, determined by trial and error mostly. We normally poured the plates and left them out (not bagged) overnight, if media are not sensitive to temperature. Plates stored in the refrigerator will require longer drying time. The spots need to dry up before the plates are inverted for incubation, normally less then 10 min. Most importantly, DO NOT flame the agar surface.
If all failed, increase the agar concentration (e.g. 17g agar, instead of 15 g, for Luria-Bertani agar), or decrease the droplet volume to 7 ul (need to modify calculation) tend to minimize spreading- particularly suited for some media containing dyes which results in more "runs". A few small batch tests should help you determine the best condition for your particular application.
Please let me know if you have more questions. Thank you for your interest in the 6x6 drop plate method.
As the above contributors indicate, drying of plates is the first step. Unfortunately if the plates are too dry the dried agar surface tension may occasionally cause your applied drop to ball-up and run. This is less common on agar media containing surfactants such as bile salts, SDS/LAS, Tween(s), Tergitol, etc. However, it can still occur if the agar surface is too dry. Drops will spread over larger areas ion surfactant-containing media as well, so one generally needs to use less dilution-associated drop-areas per-plate.
I've found plate drying highly dependent on atmospheric relative humidity. In the Antarctic with extremely dry RH it was all I could do to keep plates from drying out by gagging them tightly. In Brazil I had to leave them overnight upright with the lids on. An optimally-dried criterion I use is absence of visible precipitation on the majority of the lid-underside and on the medium surface. If putting them back, then removing them from refrigeration, I remove the plates from the sleeve ans let any condensation (again depends on atmosphere) evaporate first before using them for drop-plates. Usually by setting aside a plate from a removed batch and keeping an eye on it.
Automated-dispensing P200 pipettes are ideal for drop-plates. However, for precision you have to ensure before dispensing you remove (by touching to inside of dilution tube generally) liquid hanging onto the outside of the pipette tip after you remove it from the dilution liquid. I also generally discard the first drop. And avoid using the last drop of, for example; 20, 10 ul drops from an automated P200 into which you've loaded 200 ul. The combined dispensing imprecision of the prior drops generally leads to the last having greater variability.
One more thing occurred to me in a process of making a new batch of plates :)), if anything else does not work, you can switch to larger plates! I am working with large libraries and for my assays I have to plate different treatments and dilutions on one plate for comparison, often more than 400 spots per plate!
So, I use large plates for that (http://www.capitolscientific.com/Nunc-267060-Rectangular-1-Well-x-90mL-Cell-Culture-Dish-With-Lid-Non-Treated-Polystyrene-Sterile). There are also smaller rectangular plates (https://www.thermofisher.com/order/catalog/product/267060)
If the media continue to cause problem, you can try the adjustable spacer pipettes (https://www.shoprainin.com/Pipettes/Adjustable-Spacer-Electronic-Pipettes/c/MTEPAS001 or the manual version; other manufacturers may also have similar products) to increase the spacing between drops- great for spotting onto the square plates with grid, too.