Chain Cleaning Frequency vs Cost
The answer to the chain/drive train cleaning frequency question is pretty ambiguous. Allow your good sense to prevail as there is a fine line between meticulous and a psychiatric diagnosis. I ride daily using either mountain bike or roadie and in all weather conditions except ice or snow >6 inches. I use TriFlow (just a fact, not a product endorsement) to lube chain, derailleur pivot points, shifter &amp; brake internals.  I find that every week to 10 days, the chain requires the full cleanse and relube, which, as noted in other answers, has many routes to the goal of a lubed chain free of contaminates. 

As far as in-between these complete cleansings, here's some ideas:
--Compressed air. Clinging dirt and sand get blown away rather easily and quickly with short  blasts from the nozzle. I also hit the cassette and derailleurs.  The Teflon in my choice of lube helps matters as road grit is less likely to stick firmly, if at all, to the chain or components. This is quick and easy enough to do after every ride, though the conditions on every ride don't always create the need for cleaning. Like pressurized water, good sense and care must be used so that the contaminants aren't driven into bearing housing, shift and brake cabling, etc.  It's also possible to blow lubrication layer off the chain if one blasts a spot for too long. Overall, this works great.

--Take a rag, wrap around the chain and back peddle the crank to effectively remove much of the gunk collected on a chain. Related to this, one should wipe excess lube off the chain a short period of time after applying to chain and allowing it to penetrate the only place on the chain where it's needed: the internal aspects of the linkages where pin, bushing and side plates come together. 

-- Excess lube creates an opportunity for more dirt and grime to attach itself to the chain. When applying lube, a couple of drops on each linkage followed by normal chain movement to assist it's penetration into the linkages is ample.  Allow the solvent carrier (if applicable to your choice of lube) to evaporate and wipe off the excess clinging to the side plates. 

--I find that paying attention to the rear derailleur pulleys is a helpful indicator when assessing the contamination level. It's here on the pulleys where the gritty, chain-wearing paste accumulates and transfers to and from the chain as one rides along.  It's a simple matter to take a cloth or scraping tool and remove this from the pulley wheels. 

--Some bicyclists like to run two chains, swapping the clean one in and gaining some flexibility with the dirty one, which can be cleaned at a time that's more convenient. The advent and success of Quick-links to break and rejoin a chain multiple times make this an easy way to keep the drivetrain fresh. Many report both cassette and chain have gained excellent longevity using this technique (more than can be explained by simply doubling the typical life of a single chain). 

That was a Texas sized answer to say run two chains intermittently and utilize compressed air for quick cleaning between changes.