Daily

Scraping large particles into the bin or compost means you won't have to pick them out of the filter later, but there's no need to rinse plates before racking them (rinsing can even fool your dishwasher's automatic mode into thinking your crockery's cleaner than it is, so it won't wash as well).

Leave the door ajar. This isn't always practical, but this helps extend the life of your door seals, and circulating air helps keep mould and odours at bay. Some models even open automatically to aid drying performance.
Use good quality dishwasher detergent. Not only will they do a better job, quality cleaners are less likely to leave detergent residue behind and help prevent slimy deposits from building up in your appliance.

Weekly

Clean your dishwasher seals. Grime build-up around seals causes them to deteriorate more quickly. Run a damp cloth over door seals and contact points to keep them in top condition.
Clean the filter. Washing out the internal filter under hot running water and using a plastic brush to lightly scrub off the dirt means your dishwasher will clean more efficiently (and means less grime makes its way deeper into your appliance).

Monthly

Give the exterior a once-over. This not only keeps your kitchen looking spick and span, it helps keep corrosion at bay. Clean your dishwasher's door and fascia with a soft cloth and hot soapy water (or an all-purpose cleaner, but check your manual first), paying attention to corners, handles and around the controls.
Fill the rinse aid dispenser. Topping up the reservoir next to the detergent dispenser will help keep your glasses streak-free. Many dishwasher tablets include their own rinse aid, but it's extra important if you prefer powders.

Annually

Run the dishwasher on its hottest cycle while empty. Or better yet, pour a couple of cups of vinegar into a bowl on the bottom rack. This helps flush grease, limescale and deposits out of the pipes and removes foul odours.
Check the spray arms. Grime can become lodged inside the jets, making them less effective. Remove the spray arms and poke a toothpick into the water exit holes to clear any obstructions and keep them flowing freely.

Should I use a commercial dishwasher cleaner?

There are several commercially available cleaning products that claim to clear out your dishwasher's internals and leave it as good as new.

Running a cleaner through your dishwasher is a great way to flush away built up grease and limescale, but unless you've let neglect get the better of you and you're now dealing with a decade's worth of grime in one go, you may better off with plain old white vinegar.

An alternative to a commercial dishwasher cleaner is Vinegar. Like its commercial counterparts vinegar helps shift discolorations, foul smells, and nasties built up in your pipes, but has the added benefits of being less toxic, cheap and readily available.

How to clean your dishwasher with vinegar

Pour two cups into a bowl on the bottom rack of your otherwise empty appliance
Give it a run, pausing the cycle in the middle for half an hour or so to give it an extra good soaking.
If the dishwasher still gives olfactory offense, throw in a generous handful of baking soda and run it again to give stubborn smells their marching orders.

Why pour the vinegar into a bowl and not just tip it straight in? Because many dishwashers run their drain pump at the start of the cycle to clear residual waste water left over from the last run, and this would pump your vinegar straight down the drain. Using a bowl bypasses this initial pump out and keeps the vinegar in the machine where it's needed.

Dishwasher brand